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Two types of hypergeometric degenerate Cauchy numbers

  • Takao Komatsu EMAIL logo
Published/Copyright: May 29, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

Keywords: hypergeometric Cauchy numbers; degenerate Cauchy numbers; hypergeometric functions; zeta functions; determinants
MSC 2010: Primary: 11B75; Secondary: 11B37; 11C20; 11M41; 15A15; 33C05

1 Introduction

In 1985, Howard [1, (3.6)] introduced the polynomial β n ( k ) ( λ , x ) , defined by means of the generating function

(1) ( t ( 1 + λ t ) 1 / λ − 1 ) k ( 1 + λ t ) x / λ = ∑ n = 0 ∞ β n ( k ) ( λ , x ) t n n ! .

When k = 1, β n ( λ , x ) = β n ( 1 ) ( λ , x ) are the degenerate Bernoulli polynomials, introduced by Carlitz [2] and defined by

(2) ( t ( 1 + λ t ) 1 / λ − 1 ) ( 1 + λ t ) x / λ = ∑ n = 0 ∞ β n ( λ , x ) t n n ! .

Nowadays, the polynomials in (1) are called the degenerate Bernoulli polynomials of higher order. When x = 0 in (3), β n (λ) = β n (λ, 0) are the degenerate Bernoulli numbers, defined by

(3) t ( 1 + λ t ) 1 / λ − 1 = ∑ n = 0 ∞ β n ( λ ) t n n ! .

When λ → 0 in (2), B n (x) = β n (0, x) are the ordinary Bernoulli polynomials because

lim λ → 0 ( t ( 1 + λ t ) 1 / λ − 1 ) ( 1 + λ t ) x / λ = t e x t e t − 1 = ∑ n = 0 ∞ B n ( x ) t n n ! .

When λ → 0 and x = 0 in (2), B n = β n (0, 0) are the classical Bernoulli numbers defined by

(4) t e t − 1 = ∑ n = 0 ∞ B n t n n ! .

The degenerate Bernoulli polynomials in λ and x have rational coefficients. When x = 0, β n (λ) = β(λ, 0) are called degenerate Bernoulli numbers. In [3], explicit formulas for the coefficients of the polynomial β n (λ) are found. In [4], a general symmetric identity involving the degenerate Bernoulli polynomials and the sums of generalized falling factorials are proved. In [5], a kind of generalization of degenerate Bernoulli numbers, called hypergeometric degenerate Bernoulli numbers, is introduced and its properties are studied.

The classical Bernoulli numbers may be called Bernoulli numbers of the first kind too. It is because there exists another kind of Bernoulli numbers, that is, the Bernoulli numbers of the second kind b n , defined by

(5) x log ( 1 + x ) = ∑ n = 0 ∞ b n x n .

Bernoulli numbers of the second kind b n are often studied as Cauchy numbers (of the first kind) c n . As seen in [3, Lemma 2.1], the following relation holds.

(6) c n = n ! b n .

It is natural that Cauchy numbers are recognized by t being replaced by log(1 + t) and taking its reciprocal in the generating function in (4):

e log ( 1 + t ) − 1 log ( 1 + t ) = ∑ n = 0 ∞ λ n t n n ! .

On the contrary, Bernoulli numbers can be recognized by t being replaced by e t − 1 and taking its reciprocal in the generating function of Cauchy numbers. In fact, log(1 + t) and e t − 1 are inverse functions of each other. Therefore, it is natural to introduce the degenerate Cauchy numbers γ n (λ) by replacing t by log(1 + t) and taking its reciprocal in (2). In fact, in the same paper, as the counterpart of degenerate Bernoulli polynomial, Howard [1, (3.7)] introduces another polynomial γ n ( k ) ( λ , x ) , defined by means of the generating function

(7) ( λ t ( 1 + t ) λ − 1 ) k 1 ( 1 + t ) x − λ = ∑ n = 0 ∞ γ n ( k ) ( x | λ ) t n n ! .

This should be called the degenerate Cauchy polynomials. When k = 1, the degenerate Cauchy polynomials γ n ( x | λ ) = γ n ( 1 ) ( x | λ ) are defined by

(8) λ t ( ( 1 + t ) λ − 1 ) ( 1 + t ) x − λ = ∑ n = 0 ∞ γ n ( x | λ ) t n n ! .

When x = λ in (8), degenerate Cauchy numbers γ n (λ) = γ n (λ|λ) can be naturally defined by

(9) λ t ( 1 + t ) λ − 1 = ∑ n = 0 ∞ γ n ( λ ) t n n !

(see also [6, (2.13)]). This matches our idea about the relation between Bernoulli and Cauchy polynomials by replacing t by log(1 + t) and taking its reciprocal in (2).

It seems that this type of degenerate Cauchy polynomials has not been well studied but forgotten. Unfortunately, there seem to exist another definition of degenerate Cauchy polynomials without any historical background or motivation.

2 Degenerate Cauchy numbers

Carlitz [2] has defined the degenerate Stirling numbers of the first and second kinds, s ¯ ( n , r | λ ) and S(n, r|λ) by

(10) ( 1 − ( 1 − t ) λ λ ) r = r ! ∑ n = r ∞ s ¯ ( n , r | λ ) t n n !

and

(11) ( ( 1 + λ t ) 1 / λ − 1 ) r = r ! ∑ n = r ∞ S ( n , r | λ ) t n n ! ,

respectively. When λ → 0, [ n r ] = ( − 1 ) n − r s ¯ ( n , r | 0 ) are the ordinary (unsigned) Stirling numbers of the first kind and { n r } = S ( n , r | 0 ) are the ordinary Stirling numbers of the second kind. Their generating functions are given by

(12) ( − log ( 1 − t ) ) k k ! = ∑ n = k ∞ [ n k ] t n n !

and

(13) ( e t − 1 ) k k ! = ∑ n = k ∞ { n k } t n n ! ,

respectively.

In a similar relation between (2) and (10) with (12), from (11) with (13) the degenerate Cauchy numbers γ n (λ) can be naturally defined in (9).

Howard [3, Theorem 3.1] determined all the coefficients of β n (λ) in (3) as

β n ( λ ) = c n λ n + ∑ k = 1 n n k ( − 1 ) n − k [ n − 1 k − 1 ] B k λ n − k ( n ≥ 1 ) .

In particular, the leading coefficient is the classical Cauchy numbers c n and the constant term is the classical Bernoulli numbers B n . The list of some degenerate Bernoulli numbers is in Appendix.

The coefficients of γ n (λ) can also be determined. In fact, the coefficients appear in the reverse order of those of β n (λ). The list of some degenerate Cauchy numbers is in Appendix.

Theorem 1

For n ≥ 1,

γ n ( λ ) = c n + ∑ k = 1 n n k ( − 1 ) n − k [ n − 1 k − 1 ] B k λ k .

Proof

Put f = λ log(1 + t). Then, by (12)

∑ n = 0 ∞ γ n ( λ ) t n n ! = λ t ( 1 + t ) λ − 1 = λ t f f e f − 1 = λ t f + λ t f ∑ k = 1 ∞ B k f k k ! = t log ( 1 + t ) + λ t ∑ k = 1 ∞ B k k ! λ k − 1 ( log ( 1 + t ) ) k − 1 = ∑ n = 0 ∞ c n t n n ! + t ∑ k = 1 ∞ B k k λ k ∑ n = k − 1 ∞ ( − 1 ) n − k + 1 [ n k − 1 ] t n n ! = ∑ n = 0 ∞ c n t n n ! + ∑ k = 1 ∞ B k k λ k ∑ n = k ∞ ( − 1 ) n − k [ n − 1 k − 1 ] t n ( n − 1 ) ! = ∑ n = 0 ∞ c n t n n ! + ∑ n = 1 ∞ ( ∑ k = 1 n ( − 1 ) n − k [ n − 1 k − 1 ] B k ) λ k t n n ! . □

3 Hypergeometric degenerate Cauchy numbers

In [7,8], the hypergeometric Cauchy numbers c N,n (N ≥ 1, n ≥ 0) are introduced as

(14) 1 F 1 2 ( 1 , N ; N + 1 ; − t ) = ( − 1 ) N − 1 t N / N log ( 1 + t ) − ∑ n = 1 N − 1 ( − 1 ) n − 1 t n / n := ∑ n = 0 ∞ c N , n t n n ! ,

where

F 1 2 ( a , b ; c ; z ) = ∑ n = 0 ∞ ( a ) ( n ) ( b ) ( n ) ( c ) ( n ) z n n !

is the Gauss hypergeometric function with the rising factorial (x)(n) = x(x + 1)⋯(x + n − 1) (n ≥ 1) and (x)(0) = 1. In this paper, (x) n = x(x − 1)⋯(x − n + 1) (n ≥ 1) is the falling factorial with (x)0 = 1. Similar hypergeometric numbers are hypergeometric Bernoulli numbers [9,10,11] and hypergeometric Euler numbers [12,13].

Define the hypergeometric degenerate Cauchy numbers γ N,n (λ) by the generating function

(15) 1 F 1 2 ( 1 , N − λ ; N + 1 ; − t ) = ( λ − 1 ) N − 1 t N / N ! ( 1 + t ) λ − 1 λ − ∑ n = 1 N − 1 ( λ − 1 ) n − 1 t n / n ! := ∑ n = 0 ∞ γ N , n ( λ ) t n n ! .

We list some initial values of γ N,n (λ) (0 ≤ n ≤ 5) in Appendix. When λ → 0 in (15), c N,n = γ N,n (0) are the hypergeometric Cauchy numbers in (14). When N = 1 in (15), γ n (λ) = γ 1,n (λ) are the degenerate Cauchy numbers in (9).

Note that

(16) F 1 2 ( 1 , N − λ ; N + 1 ; − t ) = ∑ n = 0 ∞ ( N − λ ) ( n ) ( N + 1 ) ( n ) ( − t ) n = ∑ n = 0 ∞ ( λ − N ) n N ! ( N + n ) ! t n = ( ∑ n = N ∞ ( λ − 1 ) n − 1 n ! t n ) ( ( λ − 1 ) N − 1 N ! t N ) − 1 .

First, we shall show the recurrence relation of hypergeometric degenerate Cauchy numbers.

Proposition 1

For n ≥ 1 and N ≥ 1, we have

γ N , n ( λ ) = − n ! ∑ k = 0 n − 1 ( λ − N ) n − k N ! ( N + n − k ) ! k ! γ N , k ( λ )

with γ N,0(λ) = 1.

Proof

From the definition in (15),

1 = ( 1 + ∑ l = 1 ∞ ( λ − N ) l N ! ( N + l ) ! t l ) ( ∑ n = 0 ∞ γ N , n ( λ ) t n n ! ) = ∑ n = 0 ∞ γ N , n ( λ ) t n n ! + ∑ n = 1 ∞ ∑ k = 0 n − 1 ( λ − N ) n − k N ! ( N + n − k ) ! γ N , k ( λ ) k ! t n .

Comparing the coefficients on both sides, we get the desired result.□

The hypergeometric degenerate Cauchy numbers can be expressed in terms of determinants.

Theorem 2

For n, N ≥ 1,

(17) γ N , n ( λ ) = ( − 1 ) n n ! | ( λ − N ) N ! ( N + 1 ) ! 1 0 ( λ − N ) 2 N ! ( N + 2 ) ! ( λ − N ) N ! ( N + 1 ) ! ⋮ ⋮ ⋱ 1 ( λ − N ) n − 1 N ! ( N + n − 1 ) ! ( λ − N ) n − 2 N ! ( N + n − 2 ) ! ⋯ ( λ − N ) N ! ( N + 1 ) ! 1 ( λ − N ) n N ! ( N + n ) ! ( λ − N ) n − 1 N ! ( N + n − 1 ) ! ⋯ ( λ − N ) 2 N ! ( N + 2 ) ! ( λ − N ) N ! ( N + 1 ) ! | .

Remark

When λ → 0, Theorem 2 is reduced to [7, Theorem 1]. When λ → 0 and N = 1, we have a determinant expression of Cauchy numbers by Glaisher [14, p. 50].

Proof

The proof is obtained by induction on n. When n = 1, the result is trivial because γ N , 1 ( λ ) = − ( λ − N ) N ! ( N + 1 ) ! . Assume that the result is valid up to n − 1. Then, the determinant of the right-hand side (RHS) of (17) is expanded along the first row.

( λ − N ) N ! ( N + 1 ) ! ( − 1 ) n − 1 γ N , n − 1 ( γ ) ( n − 1 ) ! − | ( λ − N ) 2 N ! ( N + 2 ) ! 1 0 ( λ − N ) 3 N ! ( N + 3 ) ! ( λ − N ) N ! ( N + 1 ) ! ⋮ ⋮ ⋱ 1 ( λ − N ) n − 1 N ! ( N + n − 1 ) ! ( λ − N ) n − 3 N ! ( N + n − 3 ) ! ⋯ ( λ − N ) N ! ( N + 1 ) ! 1 ( λ − N ) n N ! ( N + n ) ! ( λ − N ) n − 2 N ! ( N + n − 2 ) ! ⋯ ( λ − N ) 2 N ! ( N + 2 ) ! ( λ − N ) N ! ( N + 1 ) ! | = … = ( λ − N ) N ! ( N + 1 ) ! ( − 1 ) n − 1 γ N , n − 1 ( γ ) ( n − 1 ) ! − ( λ − N ) 2 N ! ( N + 2 ) ! ( − 1 ) n − 2 γ N , n − 2 ( γ ) ( n − 2 ) ! + ⋯ + | ( N λ ) n − 1 N ! ( N + n − 1 ) ! 1 ( λ − N ) n N ! ( N + n ) ! ( λ − N ) N ! ( N + 1 ) ! | = ( − 1 ) n − 1 ∑ m = 0 n − 1 ( λ − N ) n − m N ! ( N + n − m ) ! γ N , m ( λ ) m ! = ( − 1 ) n γ N , n ( λ ) n ! .

Here, we used the recurrence relation in Proposition 1.□

The hypergeometric degenerate Cauchy numbers can be expressed explicitly.

Theorem 3

For n, N ≥ 1, we have

γ N , n ( λ ) = n ! ∑ k = 1 n ( − N ! ) k ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 ( λ − N ) i 1 ( N + i 1 ) ! ⋯ ( λ − N ) i k ( N + i k ) ! .

Proof

The proof is obtained by induction on n. When n = 1, the result is valid because γ N , 1 ( λ ) = − ( λ − N ) N ! ( N + 1 ) ! . Assume that the result is valid up to n − 1. Then, by using the recurrence relation in Proposition 1, we have

γ N , n ( λ ) n ! = − ∑ l = 0 n − 1 ( λ − N ) n − l N ! ( N + n − l ) ! γ N , l ( λ ) l ! = − ∑ l = 1 n − 1 ( λ − N ) n − l N ! ( N + n − l ) ! ∑ k = 1 l ( − N ! ) k ∑ i 1 + ⋯ + i k = 1 i 1 , … , i k ≥ 1 ( λ − N ) i 1 ( N + i 1 ) ! ⋯ ( λ − N ) i k ( N + i k ) ! − ( λ − N ) n N ! ( N + n ) ! = − ∑ k = 1 n − 1 ( − N ! ) k ∑ l = k n − 1 ( λ − N ) n − l N ! ( N + n − l ) ! ∑ i 1 + ⋯ + i k = 1 i 1 , … , i k ≥ 1 ( λ − N ) i 1 ( N + i 1 ) ! ⋯ ( λ − N ) i k ( N + i k ) ! − ( λ − N ) n N ! ( N + n ) ! = − ∑ k = 2 n ( − N ! ) k − 1 ∑ l = k − 1 n − 1 ( λ − N ) n − l N ! ( N + n − l ) ! ∑ i 1 + ⋯ + i k − 1 = l i 1 , … , i k − 1 ≥ 1 ( λ − N ) i 1 ( N + i 1 ) ! ⋯ ( λ − N ) i k − 1 ( N + i k − 1 ) ! − ( λ − N ) n N ! ( N + n ) ! = ∑ k = 2 n ( − N ! ) k ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 ( λ − N ) i 1 ( N + i 1 ) ! ⋯ ( λ − N ) i k ( N + i k ) ! ( i k = n − l ≥ 1 ) − ( λ − N ) n N ! ( N + n ) ! = ∑ k = 1 n ( − N ! ) k ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 ( λ − N ) i 1 ( N + i 1 ) ! ⋯ ( λ − N ) i k ( N + i k ) ! . □

There exists another form of the hypergeometric degenerate Cauchy numbers.

Theorem 4

For n, N ≥ 1, we have

γ N , n ( λ ) = n ! ∑ k = 1 n ( − N ! ) k ( n + 1 k + 1 ) ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 0 ( λ − N ) i 1 ( N + i 1 ) ! ⋯ ( λ − N ) i k ( N + i k ) ! .

Proof

The proof can be obtained similar to that of Theorem 3. However, we give a different proof here. Put 1 + w = 2 F 1(1, N − λ; N + 1; −t). By the definition in (15),

γ N , n ( λ ) = d n d t n ( 1 + w ) − 1 | t = 0 = d n d t n ( ∑ l = 0 ∞ ( − w ) l ) | t = 0 = ∑ l = 0 ∞ d n d t n ( − w ) l | t = 0 = ∑ l = 0 ∞ ∑ k = 0 l ( − 1 ) k ( l k ) d n d t n ( 2 F 1 ( 1 , N − λ ; N + 1 ; − t ) ) k | t = 0 .

By relation (16), we know that

d n d t n ( 2 F 1 ( 1 , N − λ ; N + 1 ; − t ) ) k | t = 0 = d n d t n ( ∑ j = 0 ∞ ( λ − N ) j N ! ( N + j ) ! t j ) k | t = 0 = n ! R k ( n ) ,

where

R k ( n ) = ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 0 ( λ − N ) i 1 N ! ( N + i 1 ) ! ⋯ ( λ − N ) i k N ! ( N + i k ) ! .

Since R k (n) = 0, if k = 0, we have

γ N , n ( λ ) = ∑ l =0 n ∑ k =0 l ( − 1) k ( l k ) n ! R k ( n ) = n ! ∑ k =0 n ( − 1) k ∑ l = k n ( l k ) R k ( n ) = n ! ∑ k =1 n ( − 1) k R k ( n ) ∑ l = k n ( l k ) = n ! ∑ k =1 n ( − 1) k ( n + 1 k + 1 ) R k ( n ) . □

3.1 Coefficients of the hypergeometric degenerate Cauchy numbers

Some coefficients of γ N,n (λ) can also be explicitly described. In particular, the leading coefficient is the hypergeometric Bernoulli numbers B N,n and the constant is the hypergeometric Cauchy numbers c N,n . When N = 1, this result is reduced to the coefficients given in Theorem 1.

Theorem 5

The leading coefficient of γ N,n (λ) is equal to B N,n and its constant term is equal to c N,n .

Proof

From Theorem 3, we get the leading coefficient of γ N,n (λ) with λ n as

n ! ∑ k =1 n ( − N ! ) k ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 λ i 1 ( N + i 1 ) ! ⋯ λ i k ( N + i k ) ! = n ! ∑ k =1 n ( − N ! ) k ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 1 ( N + i 1 ) ! ⋯ ( N + i k ) ! λ n ,

which is equal to the expression of B N,n  λ n in [15, Proposition 2]. From Theorem 3 again, we get the constant term of γ N,n (λ) as

n ! ∑ k = 1 n ( − N ! ) k ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 ( − N ) i 1 ( N + i 1 ) ! ⋯ ( − N ) i k ( N + i k ) ! = n ! ∑ k = 1 n ( − 1 ) n − k ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 N k ( N + i 1 ) ⋯ ( N + i k ) ,

which is equal to the expression of c N,n in [7, Lemma 1].□

Nevertheless, it seems difficult to determine other coefficients in any explicit form.

4 A different kind of hypergeometric degenerate Cauchy numbers

The Cauchy numbers c n are defined by the generating function

G C ( x ) := x log ( 1 + x ) = ∑ n = 0 ∞ c n x n n ! .

Notice that Cauchy numbers are strongly related to Bernoulli numbers B n via the relation

(18) G C ( x ) = 1 G B ( log ( 1 + x ) ) ,

where G B(x) is the generating function of Bernoulli numbers

G B ( x ) ≔ x e x − 1 = ∑ n = 0 ∞ B n x n n ! .

One kind of generalization of Bernoulli numbers is called hypergeometric Bernoulli numbers B N,n ([9,10,11]), defined by

G HB , N ( x ) ≔ 1 F 1 1 ( 1 ; N + 1 ; x ) = ∑ n = 0 ∞ B N , n x n n ! ,

where

F 1 1 ( a ; b ; z ) = ∑ n = 0 ∞ ( a ) ( n ) ( b ) ( n ) z n n !

with the rising factorial (x)(n) = x(x + 1)⋯(x + n − 1) (n ≥ 1) and (x)(0) = 1 is the confluent hypergeometric function. When N = 1, we have B n = B 1,n . One kind of generalization of Cauchy numbers is called hypergeometric Cauchy numbers c N,n ([8]). When N = 1, we have c n = c 1,n . Nevertheless, in [16], another type of hypergeometric numbers is introduced with respect to (18). For n ≥ 0 and N ≥ 1, H-Cauchy numbers are defined by

G HC , N ( x ) ≔ 1 N ! G HB , N ( log ( 1 + x ) ) = F 1 1 ( 1 ; N + 1 ; log ( 1 + x ) ) N ! = ∑ n = 0 ∞ C n ( N ) x n n ! .

In this section, similar to H-Cauchy numbers, we shall introduce a different kind of hypergeometric degenerate Cauchy numbers with respect to (18). The H-degenerate Cauchy numbers C N,n (λ) are defined by the generating function

(19) G HDC , N ( x ) ≔ 1 N ! G HDB , N ( log ( 1 + x ) ) = F 1 2 ( 1 , N − 1 / λ ; N + 1 ; − λ log ( 1 + x ) ) N ! = ∑ n = 0 ∞ C N , n ( λ ) x n n ! ,

where the hypergeometric degenerate Bernoulli numbers β N,n (γ) ([17]) are defined by

G HDB , N ( x ) ≔ 1 F 1 2 ( 1 , N − 1 / λ ; N + 1 ; − λ x ) = ∑ n = 0 ∞ β N , n ( λ ) x n n ! .

In this case, another type of degenerate Cauchy numbers C 1,n (λ) are given by

(20) ( 1 + λ log ( 1 + x ) ) 1 / λ − 1 log ( 1 + x ) = ∑ n = 0 ∞ C 1 , n ( λ ) x n n ! .

When λ → 0, the degenerate Cauchy numbers in (20) are reduced to the classical Cauchy numbers c n . We list some initial values of C N,n (λ) (0 ≤ n ≤ 5) in Appendix.

H-degenerate Cauchy numbers can be expressed explicitly in terms of the (unsigned) Stirling numbers of the first kind [ n m ] . Here, (a|b) n = a(a − b)⋯(a − (n − 1)b) (n ≥ 1) is the generalized falling factorial with (a|b)0 = 1. Note that when b = 1, (a) n = (a|1) n is the original falling factorial.

Theorem 6

For N, n ≥ 0, we have

C N , n ( λ ) = ∑ m = 0 n [ n m ] ( − 1 ) n − m ( 1 − N λ | λ ) m m ! ( N + m ) ! .

Remark

When λ → 0 in Theorem 6, we have

C N , n ( 0 ) = ∑ m = 0 n [ n m ] ( − 1 ) n − m m ! ( N + m ) ! ,

that is [16, Theorem 3.1] (see also [18, Theorem 2.2]). When N = 1 in Theorem 6, we have

C 1 , n ( λ ) = ∑ m = 0 n [ n m ] ( − 1 ) n − m ( 1 − λ | λ ) m m + 1 ,

which shows an explicit expression of another type of degenerate Cauchy numbers. When λ → 0 and N = 1 in Theorem 6, we have

C 1 , n ( 0 ) = ∑ m = 0 n [ n m ] ( − 1 ) n − m m + 1 = c n ,

that is, an explicit expression of the classical Cauchy numbers ([19, Ch. VII], [20, p. 1908]).

Proof of Theorem 6

By the definition in (19),

∑ n = 0 ∞ C N , n ( λ ) x n n ! = 1 N ! ∑ m = 0 ∞ ( 1 − N λ | λ ) m ( N + m ) m ( log ( 1 + x ) ) m = ∑ m = 0 ∞ ( 1 − N λ | λ ) m m ! ( N + m ) ! ( log ( 1 + x ) ) m m ! = ∑ m = 0 ∞ ( 1 − N λ | λ ) m m ! ( N + m ) ! ∑ n = m ∞ ( − 1 ) n − m [ n m ] x n n ! = ∑ n = 0 ∞ ∑ m = 0 n [ n m ] ( − 1 ) n − m ( 1 − N λ | λ ) m m ! ( N + m ) ! x n n ! .

Comparing the coefficients on both sides, we get the desired result.□

By Theorem 6, we obtain an expression of H-degenerate Cauchy numbers for negative indices, where the definition in (19) can be interpreted as

∑ n = 0 ∞ C − N , n ( λ ) x n n ! = ∑ n = N ∞ ( N − 1 / λ ) ( n ) ( n − N ) ! ( − λ log ( 1 + x ) ) n .

Corollary 1

For N, n ≥ 0, we have

C − N , n ( λ ) = ∑ m = N n [ n m ] ( − 1 ) n − m ( 1 + N λ | λ ) m m ! ( m − N ) ! .

Theorem 7

∑ n = 0 ∞ ∑ N = 0 ∞ C − N , n ( λ ) x n n ! y N N ! = ( 1 + λ log ( 1 + x ) ) 1 / λ ( 1 − λ y log ( 1 + x ) ) 1 + 1 / λ .

Remark

When λ → 0,

RHS → e log ( 1 + x ) 1 ⋅ e − y log ( 1 + x ) = ( 1 + x ) y ,

that is, the result in [16, Theorem 3.4].

Proof of Theorem 7

By Corollary 1 with the proof of Theorem 6, we have

LHS = ∑ N = 0 ∞ ∑ m = N ∞ ( 1 + N λ | λ ) m ( m − N ) ! ( log ( 1 + x ) ) m y N N ! = ∑ N = 0 ∞ ∑ m = 0 ∞ ( 1 + N λ | λ ) m + N m ! ( log ( 1 + x ) ) m + N y N N ! = ∑ m = 0 ∞ ( 1 / λ ) m ( λ log ( 1 + x ) ) m m ! ∑ N = 0 ∞ ( N + 1 / λ ) N ( λ y log ( 1 + x ) ) N N ! = ∑ m = 0 ∞ ( 1 / λ m ) ( λ log ( 1 + x ) ) m ∑ N = 0 ∞ ( N + 1 / λ N ) ( λ y log ( 1 + x ) ) N = ∑ m = 0 ∞ ( 1 / λ m ) ( λ log ( 1 + x ) ) m ∑ N = 0 ∞ ( − 1 − 1 / λ N ) ( − λ y log ( 1 + x ) ) N = ( 1 + λ log ( 1 + x ) ) 1 / λ ( 1 − λ y log ( 1 + x ) ) − 1 − 1 / λ = RHS . □

5 Hypergeometric degenerate Cauchy polynomials

The H-degenerate Cauchy polynomials C N,n (λ, w; z) are defined by the generating function

(21) ( 1 + x ) w z F 1 2 ( 1 , N − 1 / λ ; N + 1 ; w λ log ( 1 + x ) ) N ! = ( 1 + x ) w z ∑ n = 0 ∞ ( 1 − N λ | λ ) n ( N + n ) ! ( − w log ( 1 + x ) ) n = ∑ n = 0 ∞ C N , n ( λ , w ; z ) x n n ! .

When w = −1, C N,n (λ, −1; z) are the H-degenerate Cauchy polynomials of the first kind. When w = 1, C N,n (λ, 1; z) are the H-degenerate Cauchy polynomials of the second kind. When λ → 0, C N,n (0, −1; z) and C N,n (0, −1; z) are the H-Cauchy polynomials of the first kind and those of the second kind, respectively [16]. Furthermore, when N = 1 and z = 0, c n = C 1,n (0, −1; 0) and c ˆ n = C 1 , n ( 0 , − 1 ; 0 ) are the Cauchy polynomials of the first kind and those of the second kind, respectively [19,20]. Note that

x ( 1 + x ) log ( 1 + x ) = ∑ n = 0 ∞ c ˆ n x n n ! .

First, we shall show an explicit expression of the H-degenerate Cauchy polynomials.

Theorem 8

For N, n ≥ 0, we have

C N , n ( λ , w ; z ) = ∑ m = 0 n [ n m ] ( − 1 ) n − m ( − w ) m ∑ j = 0 m ( m j ) ( 1 − N λ | λ ) m − j ( m − j ) ! ( N + m − j ) ! ( − z ) j .

Proof

By the definition in (21) with the proof of Theorem 6, we have

∑ n = 0 ∞ C N , n ( λ , w ; z ) x n n ! = ∑ j = 0 ∞ ( w z log ( 1 + x ) ) j j ! ∑ l = 0 ∞ ( 1 − N λ | λ ) l ( − w log ( 1 + x ) ) l ( N + l ) ! = ∑ m = 0 ∞ ( w log ( 1 + x ) ) m m ! ∑ j = 0 m ( − 1 ) m − j z j ( 1 − N λ | λ ) m − j m ! j ! ( N + m − j ) ! = ∑ n = 0 ∞ ∑ m = 0 n [ n m ] ( − 1 ) n − m ( − w ) m ∑ j = 0 m ( m j ) ( 1 − N λ | λ ) m − j ( m − j ) ! ( N + m − j ) ! ( − z ) j x n n ! .

Comparing the coefficients on both sides, we get the desired result.□

If one takes the summation in terms of the Stirling numbers of the second kind, we have the following identity.

Theorem 9

For N, n ≥ 1, we have

∑ m = 0 n { n m } C N , n ( λ , w ; z ) = ( − w ) m ∑ j = 0 n ( n j ) ( 1 − N λ | λ ) n − j ( n − j ) ! ( N + n − j ) ! ( − z ) j .

Proof

We use the orthogonal relation between two kinds of Stirling numbers:

∑ m = l n ( − 1) m − l [ m l ] { n m } = { 1 ( n = l ) , 0 ( n ≠ l ) .

By Theorem 8, we have

∑ m = 0 n { n m } C N , n ( λ , w ; z ) = ∑ m = 0 n { n m } ∑ l = 0 m [ m l ] ( − 1 ) m − l ( − w ) l ∑ j = 0 l ( l j ) ( 1 − N λ | λ ) l − j ( l − j ) ! ( N + l − j ) ! ( − z ) j = ∑ l = 0 n ∑ m = l n { n m } [ m l ] ( − 1 ) m − l ( − w ) l ∑ j = 0 l ( l j ) ( 1 − N λ | λ ) l − j ( l − j ) ! ( N + l − j ) ! ( − z ) j = ( − w ) n ∑ j =0 n ( n j ) (1 − N λ | λ ) n − j ( n − j ) ! ( N + n − j ) ! ( − z ) j . □

5.1 Relations between two kinds of H-degenerate Cauchy polynomials

There are two kinds of H-degenerate Cauchy polynomials C N,n (λ, w; z) and C N,n (λ, −w; z). We shall show a relation between two kinds.

Theorem 10

( − 1 ) n C N , n ( λ , w ; z ) n ! = ∑ m = 0 n ( n − 1 m − 1 ) C N , n ( λ , − w ; z ) m ! .

Proof

First,

∑ n = 0 ∞ ( − 1 ) n C N , n ( λ , w ; z ) n ! x n = ( 1 − x ) w z ∑ m = 0 ∞ ( 1 − N λ | λ ) m ( N + m ) ! ( − w log ( 1 − x ) ) m = ( 1 + x 1 − x ) − w z ∑ m = 0 ∞ ( 1 − N λ | λ ) m ( N + m ) ! ( w log ( 1 + x 1 − x ) ) m = ∑ m = 0 ∞ C N , n ( λ , − w ; z ) ( x 1 − x ) m m ! .

Second,

∑ n = 0 ∞ ∑ m = 0 n ( n − 1 m − 1 ) C N , n ( λ , − w ; z ) m ! x n = ∑ m = 0 ∞ ∑ n = m ∞ ( n − 1 m − 1 ) x n ⋅ C N , m ( λ , − w ; z ) m ! = ∑ m = 0 ∞ ( x 1 − x ) m C N , m ( λ , − w ; z ) m ! . □

5.2 H-degenerate Cauchy polynomials with negative indices

By Theorem 8, we obtain an expression of H-degenerate Cauchy numbers with negative indices, where the definition in (21) can be interpreted as

∑ n = 0 ∞ C − N , n ( λ , w ; z ) x n n ! = ( 1 + x ) w z ∑ n = N ∞ ( 1 − N λ | λ ) n ( n − N ) ! ( − w log ( 1 + x ) ) n .

Proposition 2

For N, n ≥ 0, we have

C − N , n ( λ , w ; z ) = ∑ m = 0 n [ n m ] ( − 1 ) n − m ( − w ) m ∑ j = 0 m ( m j ) ( 1 + N λ | λ ) m − j ( m − j ) ! ( m − j − N ) ! ( − z ) j .

By Proposition 2, we have the identity for the double summations.

Theorem 11

∑ n = 0 ∞ ∑ N = 0 ∞ C − N , n ( λ ) x n n ! y N N ! = ( 1 + x ) w z ( 1 − w λ log ( 1 + x ) ) 1 / λ ( 1 + w λ y log ( 1 + x ) ) 1 + 1 / λ .

Remark

When λ → 0, RHS → (1 + x)(z−y−1)w . This is reduced to Theorem 6.2 in [16].

Proof of Theorem 11

Similar to the proof of Theorem 7, by Proposition 2,

∑ n = 0 ∞ ∑ N = 0 ∞ C − N , n ( λ ) x n n ! y N N ! = ( 1 + x ) w z ∑ N = 0 ∞ ∑ m = N ∞ ( 1 + N λ | λ ) m ( m − N ) ! ( − w log ( 1 + x ) ) m y N N ! = ( 1 + x ) w z ∑ m = 0 ∞ ( 1 / λ ) m ( − w λ log ( 1 + x ) ) m m ! ∑ N = 0 ∞ ( N + 1 / λ ) N ( − w λ y log ( 1 + x ) ) N N ! = ( 1 + x ) w z ( 1 − w λ log ( 1 + x ) ) 1 / λ ( 1 + w λ y log ( 1 + x ) ) − 1 − 1 / λ . □

5.3 Zeta functions interpolating H-degenerate Cauchy polynomials

Let 0 < c < 1. For a positive integer N, define the function Z N (s, λ, w, z) ( s , w , z ∈ ℂ ) with ℜ s > 0 and |w| < 1/c as

(22) Z N ( s , λ , w , z ) = 1 Γ ( s ) ∫ 0 1 − e − c / λ (1 − t ) w z N ! F 1 2 (1, N − 1/ λ ; N + 1; w λ log (1 − t ) ) t s − 1 d t ,

or by the change of the variables t = 1 − e −u ,

(23) Z N ( s , λ , w , z ) = 1 Γ ( s ) ∫ 0 c / λ (1 − e − u ) s − 1 N ! F 1 2 (1, N − 1/ λ ; N + 1; − w λ u ) e − (1 + w z ) u d u .

Since 2 F 1 in the integrand of (23) is continuous for |wλu| ≤ |w|c < 1, (23) is integrable.

Notice that when λ → 0, (22) and (23) are reduced to the zeta functions [16].

Z N ( s , λ , w , z ) = 1 Γ ( s ) ∫ 0 1 ( 1 − t ) w z N ! F 1 2 ( 1 , N ; N + 1 ; − w λ log ( 1 − t ) ) t s − 1 d t

and

Z N ( s , λ , w , z ) = 1 Γ ( s ) ∫ 0 ∞ ( 1 − e − u ) s − 1 N ! F 1 2 ( 1 , N ; N + 1 ; w λ u ) e − ( 1 + w z ) u d u ,

respectively.

Theorem 12

For n ≥ 0,

Z N ( − n , λ , w , z ) = C N , n ( λ , w ; z ) .

Proof

Assume that ℜ s > 0 . Rewrite (22) with Hankel contour H 1, where H 1 starts at 1 − e −c/λ and goes to δ > 0, and goes around the origin with radius δ, and goes back to 1 − e −c/λ . Since the integrand except for t s−1 is holomorphic in the neighborhood of t = 0, we have

Z N ( s , λ , w , z ) = 1 Γ ( s ) ( e 2 π i s − 1 ) ∫ H 1 ( 1 − t ) w z N ! F 1 2 ( 1 , N − 1 / λ ; N + 1 ; w λ log ( 1 − t ) ) t s − 1 d t .

Then, the RHS gives the meromorphic continuation on the whole space ℂ in s. By (21), we have Z N (−n, λ, w, z) = C N,n (λ, w; z).□

The function Z N (n, λ, w, z) can be expressed as the following form.

Theorem 13

For positive integers k and n, if | w z | < 1 − | w | c 1 + | w | c ( < 1 ) and ℜ λ > 0 , then

Z N ( n , λ , w , z ) = ∑ j = 0 ∞ ∑ m = j ∞ ( 1 − N λ | λ ) m − j m ! ( − 1 ) j n ! j ! ( N + m − j ) ! ∑ l = 1 n ( n l ) ( − 1 ) l − 1 l m w m z j P m − j ( c ( l + w z ) / λ ) ,

where

P m ( a ) = 1 − e − a ∑ k = 0 m a k k ! .

Proof. When | w z | < 1 − | w | c 1 + | w | c ( < 1 ) and ℜ λ > 0 , we get

(24) Z N ( n , λ , w , z ) = 1 N ! ( n − 1 ) ! ∫ 0 c / λ ( 1 − e − u ) n − 1 F 1 2 ( 1 , N − 1 / λ ; N + 1 ; − w λ u ) e − ( 1 + w z ) u d u = 1 ( n − 1 ) ! ∑ l = 0 n − 1 ( n − 1 l ) ( − 1 ) l ∑ m = 0 ∞ ( 1 − N λ | λ ) m w m ( N + m ) ! ∫ 0 c / λ e − ( l + 1 + w z ) u u m d u = 1 ( n − 1 ) ! ∑ l = 0 n − 1 ( n − 1 l ) ( − 1 ) l ∑ m = 0 ∞ ( 1 − N λ | λ ) m w m ( N + m ) ! ( λ + 1 + w z ) m + 1 ∫ 0 c ( l + 1 + w z ) / λ e − u u m d u .

Here, the integral and the summation in (24) are interchangeable. It is because by replacing the variable with u = v/λ,

(25) ∑ m = 0 ∞ ( 1 / λ − N ) m w m ( N + 1 ) ( m ) ∫ 0 c e − ( l + 1 + w z ) v / λ v m d v .

Its absolute value is bounded by

(26) ∫ 0 c e − ℜ ( l + 1 + w z ) v / λ ∑ m = 0 ∞ ( N + 1 / | λ | ) ( m ) ( N + 1 ) ( m ) ( | w | v ) m d v .

The series in the integrand is 2 F 1(1, N + 1/|λ|; N + 1; |w|v). Since |w|v ≤ |w|c < 1, (26) is finite. Hence, the integral and the summation in (24) are interchangeable.

Now, since

∫ 0 a e − u u m d u = m ! P m ( a ) ,

we have

(27) Z N ( n , λ , w , z ) = 1 ( n − 1 ) ! ∑ l = 0 n − 1 ( n − 1 l ) ( − 1 ) l ∑ m = 0 ∞ m ! P m ( c ( l + 1 + w z ) / λ ) ( 1 − N λ | λ ) m w m ( N + m ) ! ( l + 1 + w z ) m + 1 = 1 ( n − 1 ) ! ∑ m = 0 ∞ m ! ( 1 − N λ | λ ) m w m ( N + m ) ! ∑ l = 0 n − 1 ( n − 1 l ) ( − 1 ) l ( l + 1 ) m + 1 × ∑ j = 0 ∞ ( m + j m ) ( − w z l + 1 ) j P m ( c ( l + 1 + w z ) / λ ) = 1 n ! ∑ j = 0 ∞ ∑ m = 0 ∞ m ! ( 1 − N λ | λ ) m w m ( N + m ) ! ( m + j m ) ∑ l = 1 n ( n l ) ( − 1 ) l − 1 l m + j ( − w z ) j P m ( c ( l + w z ) / λ ) = ∑ j = 0 ∞ ∑ m = 0 ∞ ( 1 − N λ | λ ) m ( m + j ) ! ( − 1 ) j n ! j ! ( N + m ) ! ∑ l = 1 n ( n l ) ( − 1 ) l − 1 l m + j w m + j z j P m ( c ( l + w z ) / λ ) .

Here, the summations in (27) are interchangeable. It is because by

∑ m = 0 ∞ ( 1 / λ − N ) m ( λ w ) m ( N + 1 ) ( m ) 1 ( l + 1 ) m + 1 ∑ j = 0 ∞ ( m + j m ) ( − w z l + 1 ) j P m ( c ( l + 1 + w z ) / λ ) ,

its absolute value is bounded by the following.

(28) ∑ m = 0 ∞ ( N + 1 / | λ | ) ( m ) | λ w | m ( N + 1 ) ( m ) | P m ( c ( l + 1 + w z ) / λ ) | ( l + 1 − | w z | ) m + 1 = ∑ m = 0 ∞ ( N + 1 / | λ | ) ( m ) | λ w | m ( N + 1 ) ( m ) 1 ( l + 1 − | w z | ) m + 1 × | ( l + 1 + w z λ ) m + 1 ∫ 0 c e − ( l + 1 + w z ) v / λ v m d v | ≤ 1 | λ | ∑ m = 0 ∞ ( N + 1 / | λ | ) ( m ) | w | m ( N + 1 ) ( m ) | l + 1 + w z | m + 1 ( l + 1 − | w z | ) m + 1 × ∫ 0 c e − ℜ ( l + 1 + w z ) v / λ v m d v .

When | w z | < 1 − | w | c 1 + | w | c , by | w z | < ( l + 1 ) 1 − | w | c 1 + | w | c , we get

| w | v l + 1 + | w z | l + 1 − | w z | ≤ | w | c l + 1 + | w z | l + 1 − | w z | < 1 ,

so (28) converges, hence the summation and the integral are interchangeable.

Remark

When z = 0, we have

Z N ( n , λ , w , 0 ) = ∑ m = 0 ∞ m ! ( 1 − N λ | λ ) m w m n ! ( m + N ) ! ∑ l = 1 n ( n l ) ( − 1 ) l − 1 l m P m ( c l / λ ) .

When λ → 0, we have P m (c(l + 1 + wz)/λ) → 1 and hence

Z N ( n , λ , w , z ) → ∑ j = 0 ∞ ∑ m = j ∞ ( 1 − N λ | λ ) m − j m ! ( − 1 ) j n ! j ! ( N + m − j ) ! ζ n ☆ ( { 1 } m ) w m z j ,

which reduces the result in [16]. Here,

ζ n ☆ ( { 1 } r ) ≔ { ∑ 1 ≤ k 1 ≤ ⋯ ≤ k r ≤ n 1 k 1 ⋯ k r ( r ≥ 1 ) , 0 ( r = 0 ) = ∑ l = 1 n ( n l ) ( − 1 ) l − 1 l r ( r ≥ 0 )

[21, (2)].

We give an expression in terms of the Bernoulli polynomials of higher order B n ( k ) ( z ) , defined by

( x e x − 1 ) k e x z = ∑ n = 0 ∞ B n ( k ) ( z ) x n n ! .

When z = 0, B n ( k ) = B n ( k ) ( 0 ) are the Bernoulli numbers of higher order.

Theorem 14

For n ≥ 0,

Z N ( − n , λ , w , z ) = C N , n ( λ , w ; z ) = ( − 1 ) n n ! N ! ∑ i = 0 n B i ( n + 1 ) ( n − w z ) i ! ( 1 − N λ | λ ) n − i w n − i ( n − i + N ) ! .

Proof

Assume that ℜ s > 0 . Write (23) with the Hankel contour H, where H stars at + ∞ and goes to δ > 0, and goes around the origin with radius δ, and goes back to + ∞.

Z N ( s , λ , w , z ) = 1 Γ ( s ) ( e 2 π i s − 1 ) ∫ H ( 1 − e − u ) s − 1 N ! 2 F 1 ( 1 , N − 1 / λ ; N + 1 ; − w λ u ) e − ( 1 + w z ) u d u .

This expression gives the meromorphic continuation on the whole space ℂ in s. For a negative integer n,

∫ | u | = ε ( 1 − e − u ) − n − 1 N ! F 1 2 ( 1 , N − 1 / λ ; N + 1 ; − w λ u ) e − ( 1 + w z ) u d u = 2 π i   Res u = 0 1 N ! ( e u e u − 1 ) n + 1 ∑ l = 0 ∞ ( 1 − N λ | λ ) l ( w u ) l ( N + l ) ! e − ( 1 + w z ) u .

Therefore,

Z N ( − n , λ , w , z ) = ( − 1 ) n n ! N ! Res u = 0 ( e u e u − 1 ) n + 1 ∑ l = 0 ∞ ( 1 − N λ | λ ) l ( w u ) l ( N + l ) ! e − ( 1 + w z ) u = ( − 1 ) n n ! N ! Res u = 0 ( u e u − 1 ) n + 1 e ( n − w z ) u u n + 1 ∑ l = 0 ∞ ( 1 − N λ | λ ) l ( w u ) l ( N + l ) ! = ( − 1 ) n n ! N ! Res u = 0 ( ∑ m = 0 ∞ u m m ! B m ( n + 1 ) ( n − w z ) ) ( ∑ l = 0 ∞ ( 1 − N λ | λ ) l ( w u ) l ( N + l ) ! ) 1 u n + 1 = ( − 1 ) n n ! N ! Res u = 0 ∑ m = 0 ∞ ( ∑ i = 0 m B i ( n + 1 ) ( n − w z ) i ! ( 1 − N λ | λ ) m − i w m − i ( m − i + N ) ! ) u m − n − 1 = ( − 1 ) n n ! N ! ∑ i = 0 n B i ( n + 1 ) ( n − w z ) i ! ( 1 − N λ | λ ) n − i w n − i ( n − i + N ) ! . □

Acknowledgments

The author is grateful of Professor Yasushi Komori for providing his paper with helpful advice. The author thanks the anonymous referees for careful reading of the manuscript and useful suggestions.

Appendix

Bernoulli numbers B n in (4)

B 0 = 1 , B 1 = − 1 2 , B 2 = 1 6 , B 3 = 0 , B 4 = − 1 30 , B 5 = 0 , B 6 = 1 42 , B 7 = 0 , B 8 = − 1 30 , B 9 = 0 , B 10 = 5 66

Cauchy numbers c n in (5) with (6)

c 0 = 1 , c 1 = 1 2 , c 2 = − 1 6 , c 3 = 1 4 , c 4 = − 19 30 , c 5 = 9 4 , c 6 = − 863 84 , c 7 = 1 , 375 24 , c 8 = − 33 , 953 90 , c 9 = 57 , 281 20 , c 10 = − 32 , 50 , 433 132

Degenerate Bernoulli numbers β n ≔ β n (λ) in (3)

β 0 = 1 , β 1 = λ 2 − 1 2 , β 2 = − λ 2 6 + 1 6 , β 3 = λ 3 4 − λ 4 , β 4 = − 19 λ 4 720 + 2 λ 2 3 − 1 30 ,   β 5 = 9 λ 5 4 − 5 λ 3 2 + λ 4 , β 6 = − 863 λ 6 84 + 12 λ 4 − 7 λ 2 4 + 1 42 , β 7 = 1 , 375 λ 7 24 − 70 λ 5 + 105 λ 3 8 − 5 λ 12 , β 8 = − 33 , 953 λ 8 90 + 480 λ 6 − 1 , 624 λ 4 15 + 50 λ 2 9 − 1 30 , β 9 = 57 , 281 λ 9 20 − 3 , 780 λ 7 + 9 , 849 λ 5 10 − 70 λ 3 + 21 λ 20 , β 10 = − 32 , 50 , 433 λ 10 132 + 33 , 600 λ 8 − 29 , 531 λ 6 3 + 5 , 345 λ 4 6 − 91 λ 2 4 + 5 66

Degenerate Cauchy numbers γ n ≔ γ n (λ) in (9)

γ 0 = 1 , γ 1 = − λ 2 + 1 2 , γ 2 = λ 2 6 − 1 6 , γ 3 = − λ 2 4 + 1 4 , γ 4 = − λ 4 30 + 2 λ 2 3 − 19 30 , γ 5 = λ 4 4 − 5 λ 2 2 + 9 4 , γ 6 = λ 6 42 − 7 λ 4 4 + 12 λ 2 − 863 84 , γ 7 = − 5 λ 6 12 + 105 λ 4 8 − 70 λ 2 + 1 , 375 24 , γ 8 = − λ 8 30 + 50 λ 6 9 − 1 , 624 λ 4 15 + 480 λ 2 − 33 , 953 90 , γ 9 = 21 λ 8 20 − 70 λ 6 + 9 , 849 λ 4 10 − 3780 λ 2 + 57281 20 , γ 10 = 5 λ 10 66 − 91 λ 8 4 + 5 , 345 λ 6 6 − 29 , 531 λ 4 3 + 33 , 600 λ 2 − 32 , 50 , 433 132

Hypergeometric degenerate Cauchy numbers γ N,n ≔ γ N,n (λ) in (15)

γ N , 0 = 1 , γ N , 1 = − λ − N N + 1 , γ N , 2 = 2 ( λ 2 − ( N − 1 ) λ − N ) ( N + 1 ) 2 ( N + 2 ) , γ N , 3 = 3 ! ( ( N − 1 ) λ 3 − ( 2 N 2 − N + 3 ) λ 2 + ( N 2 + 2 ) ( N − 1 ) λ + N ( N 2 + N + 2 ) ) ( N + 1 ) 2 ( N + 2 ) ( N + 3 ) , γ N , 4 = 4 ! ( ( N 3 − N 2 − 6 N + 2 ) λ 4 − ( N − 1 ) ( 3 N 3 + 5 N 2 + 2 N + 12 ) λ 3 + ( 3 N 5 + 6 N 4 + 13 N 3 + 30 N 2 + 6 M + 22 ) λ 2 − ( N − 1 ) ( N 5 + 3 N 4 + 8 N 3 + 16 N 2 + 10 N + 12 ) λ − N ( N 5 + 5 N 4 + 14 N 3 + 24 N 2 + 20 N + 12 ) ) × ( ( N + 1 ) 4 ( N + 2 ) 2 ( N + 3 ) ( N + 4 ) ) − 1 , γ N , 5 = ( ( N − 1 ) ( N 3 − 3 N 2 − 14 N + 2 ) λ 5 − ( 4 N 5 − 5 N 4 − 28 N 3 + 9 N 2 − 72 N + 20 ) λ 4 + 2 ( N − 1 ) ( 3 N 5 + 7 N 4 + 11 N 3 + 37 N 2 + 5 N + 35 ) λ 3 − 2 ( 2 N 7 + 7 N 6 + 24 N 5 + 65 N 4 + 64 N 3 + 118 N 2 + 30 N + 50 ) λ 2 + ( N − 1 ) ( N 7 + 5 N 6 + 20 N 5 + 56 N 4 + 82 N 3 + 118 N 2 + 64 N + 48 ) λ + N ( N 7 + 8 N 6 + 35 N 5 + 96 N 4 + 160 N 3 + 184 N 2 + 116 N + 48 ) ) × ( ( N + 1 ) 5 ( N + 2 ) 2 ( N + 3 ) ( N + 4 ) ( N + 5 ) ) − 1

H-degenerate Cauchy numbers C N,n ≔ C N,n (λ) in (19)

C N , 0 = 1 , C N , 1 = − N λ − 1 N + 1 , C N , 2 = ( N λ − 1 ) ( 2 ( N + 1 ) λ + N ) ( N + 1 ) ( N + 2 ) , C N , 3 = 2 ( N λ − 1 ) ( 3 ( N + 1 ) λ 2 + 3 N λ + N ) ( N + 1 ) ( N + 3 ) , C N , 4 = 2 ( N λ − 1 ) ( 12 ( N + 1 ) ( N + 2 ) ( N + 3 ) λ 3 + 6 ( 3 N 3 + 15 N 2 + 18 N + 2 ) λ 2 + ( 11 N 3 + 52 N 2 + 47 N − 12 ) λ + N ( 3 N 2 + 16 N + 19 ) ) × ( ( N + 1 ) ( N + 2 ) ( N + 3 ) ( N + 4 ) ) − 1 , C N , 5 = 2 ( N λ − 1 ) ( 60 ( N + 1 ) ( N + 2 ) ( N + 3 ) ( N + 4 ) λ 4 + 120 ( N 4 + 9 N 3 + 26 N 2 + 26 N + 5 ) λ 3 + 15 ( 7 N 4 + 60 N 3 + 151 N 2 + 98 N − 20 ) λ 2 + 5 ( 10 N 4 + 88 N 3 + 221 N 2 + 119 N − 60 ) λ + N ( 12 N 3 + 118 N 2 + 357 N + 323 ) ) × ( ( N + 1 ) ( N + 2 ) ( N + 3 ) ( N + 4 ) ( N + 5 ) ) − 1

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Received: 2019-05-18
Revised: 2019-12-23
Accepted: 2020-02-18
Published Online: 2020-05-29

© 2020 Takao Komatsu, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0030
Keywords for this article
hypergeometric Cauchy numbers; degenerate Cauchy numbers; hypergeometric functions; zeta functions; determinants
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  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
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  5. The non-negative spectrum of a digraph
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  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
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  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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