λ-q-Sheffer sequence and its applications

1 Introduction

In the 1970s, Rota and Roman [1,2] began to construct a rigorous foundation for the classical umbral calculus, which was based on modern concepts like a linear functional, a linear operator, an adjoint, and so on. In the 1980s, Ihrig and Ismail [3] and Roman [4] considered the q-umbral calculus as an extended version of the classical umbral calculus. The q-Sheffer sequences occupy the central position in the theory and are characterized by the generating functions as in [1,4,5], where the q-exponential function enters. In addition, Carlitz [6,7] extended the classical Bernoulli numbers and polynomials, introducing them as q-Bernoulli and q-Euler numbers and polynomials. Since that time, several authors have studied these and other related subjects. For more details on q-umbral calculus and its applications, we let the reader refer to [3,4,6, 7,8,9, 10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27]. Recently, many mathematicians have worked on degenerate versions of special polynomials and numbers, initiated by Carlitz [28], which include the degenerate Stirling numbers of the first and second kinds, the degenerate Bernstein polynomials, the degenerate Bell numbers and polynomials, the degenerate gamma function, the degenerate gamma random variables, and so on [19,28,29, 30,31,32, 33,34,35, 36,37,38]. Furthermore, Kim and Kim [30] introduced the degenerate Sheffer sequence and λ -Sheffer sequence. Since then, some scholars have been studying many interesting identities by using them.

In view of recent interesting results in degenerate umbral calculus, the focus of this article is to study λ -q-Sheffer sequence and the degenerate q-Sheffer sequence. The outline of this article is as follows. First, we recall several definitions and very basics about q-umbral calculus, including linear functionals, differential operators, and q-Sheffer sequences. In Section 2, we will define and briefly study the properties of degenerate q-exponential functions, degenerate q-Bernoulli polynomials of order r (higher-order degenerate q-Bernoulli polynomials), and degenerate q-Euler polynomials of order r (higher-order degenerate q-Euler polynomials) ( r ∈ N ) to apply to our results in section 3. In Section 3, we introduce a family of λ -q-linear functionals on the space of polynomials and a family of λ -q-differential operators adapted to the family of λ -q-linear functionals. We also prove several results on these functionals and those λ -q-differential operators. In addition, we introduce λ -q-Sheffer polynomials and provide some examples of such polynomials. We mention that the family { s n , q ( x ∣ λ ) } λ ∈ R ∗ of λ -q-Sheffer sequences s n , q ( x ∣ λ ) are called the degenerate sequences for the Sheffer polynomial s n , q ( x ) , where lim λ → 0 s n , q ( x ∣ λ ) = s n , q ( x ) . We also explore the formula expressing one λ -q-Sheffer polynomial in terms of another λ -q-Sheffer polynomial. We study some identities arising from λ -q-Sheffer sequences. In Section 4, as applications of Section 3, we investigate some properties of degenerate q-Bernoulli polynomials of order r and degenerate q-Euler polynomials of order r , respectively, arising from λ -q-umbral calculus. We find several formulas for expressing any polynomials as a linear combination of degenerate q-Bernoulli polynomials and degenerate q-Euler polynomials with explicit coefficients, respectively. Also we establish some connections between degenerate q-Bernoulli polynomials (degenerate q-Euler polynomials) and higher-order degenerate q-Bernoulli polynomials (higher-order degenerate q-Euler polynomials).

Let q be a fixed real number between 0 and 1. The q-analogue of n ( ∈ N ) is given by

The q-binomial coefficients is

for n , k ∈ N ∪ { 0 } with k ≤ n , where [ n ] q ! = [ n ] q [ n − 1 ] q ⋯ [ 2 ] q [ 1 ] q and [ 0 ] q ! = 1 .

The q-binomial formulas are expressed as follows:

The q-exponential functions are given by

Note that e q ( t ) → e t as q → 1 .

The q-Bernoulli polynomials, which were studied by Kupershmidt [21], are given by the generating function

When x = 0 , B n , q ( 0 ) = B n , q is called the n th q-Bernoulli number.

In [17], Kim considered the q-Euler polynomials given by the generating function:

For x = 0 , E n , q = E n , q ( 0 ) , ( n ≥ 0 ) , which are called the q-Euler numbers.

For any nonzero λ ∈ R , the degenerate exponential function is defined by

By Taylor expansion, we obtain

where ( x ) 0 , λ = 1 , ( x ) n , λ = x ( x − λ ) ( x − 2 λ ) ⋯ ( x − ( n − 1 ) λ ) , ( n ≥ 1 ) . Kim et al. introduced the λ -binomial coefficients given by

From (9), we obtain

For n ≥ 0 , it is well known that the Stirling numbers of the first and second kind, respectively, are given by

and

where ( x ) n = x ( x − 1 ) … ( x − n + 1 ) , ( n ≥ 1 ) and ( x ) 0 = 1 .

Moreover, the degenerate Stirling numbers of the first and second kind, respectively, are given by

and

Now, we review briefly the fundamental properties of q-umbral calculus [14]. Let C be the complex number field, and let F be the set of all formal power series in t over C with

Let P = C [ t ] and let P ∗ be the vector space of all linear functionals on P . We denote by ⟨ L ∣ p ( x ) ⟩ the action of the linear functional L on the polynomial p ( x ) .

The vector space operations on P ∗ are given as follows:

where c is any constant in C (see [25]). For f ( t ) = ∑ k = 0 ∞ a k [ k ] q ! t k ∈ F , the linear functional on P is given by

From (17), we obtain

where δ n , k is the Kronecker’s symbol.

Let f L ( t ) = ∑ k = 0 ∞ ⟨ L ∣ x k ⟩ [ k ] q ! t k . Then, from (17) and (18), we have ⟨ f L ( t ) ∣ x n ⟩ = ⟨ L ∣ x n ⟩ and so L = f L ( t ) . Furthermore, the map L → f L ( t ) is a vector space isomorphism from P ∗ onto F . Thus, F denotes both the algebra of formal power series in t and the vector space of all linear functionals on P , and so an element f ( t ) of F is thought of as both a formal power series and a linear functional. We call F the q-umbral algebra and the q-umbral calculus is the study of q-umbral algebra.

From (4) and (17), we easily see that

The order o ( f ( t ) ) of a power series f ( t ) is the smallest integer k for which the coefficient of t k does not vanish. If o ( f ( t ) ) = 1 , then f ( t ) is called a delta series (see [1,4,37]). Notice that for all f ( t ) in F

and for all polynomials p ( x ) ,

For f 1 ( t ) , f 2 ( t ) , … , f n ( t ) ∈ F , we have

where n i 1 , … , i m q = [ n ] q ! [ i 1 ] q ! ⋯ [ i m ] q ! .

Let f ( t ) , g ( t ) ∈ F with o ( f ( t ) ) = 1 and o ( g ( t ) ) = 0 . Then there exists a unique sequence s n , q ( x ) ( deg s n , q ( x ) = n ) of polynomials such that

which is denoted by s n , q ( x ) ∼ ( g ( t ) , f ( t ) ) .

The sequence s n , q ( x ) is called the q-Sheffer sequence for ( g ( t ) , f ( t ) ) . Let s n , q ( x ) ∼ ( g ( t ) , f ( t ) ) . Then for any h ( t ) in F and for any polynomial p ( x ) , we have

and

Moreover, s n , q ( x ) is the q-Sheffer sequence for ( g ( t ) , f ( t ) ) if and only if

for all x ∈ C , where f ¯ ( f ( t ) ) = f ( f ¯ ( t ) ) = t .

2 Degenerate q-Bernoulli and degenerate q-Euler polynomials

In this section, we define the degenerate q-exponential functions, the degenerate q-Bernoulli numbers and polynomials of order r ∈ N , and the degenerate q-Euler numbers and polynomials of order r ∈ N .

In view of (9), we consider the degenerate q-binomial coefficients that are defined as follows:

From (9) and (27), we observe

From (10) and (28), we observe that

The Lah numbers are used to count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets and given by

Zou [27] considered the q-Lah numbers which are defined as

Let ( 1 − t ) q k = ∏ l = 0 k − 1 ( 1 − q l t ) = ( 1 − t ) ( 1 − q t ) ⋯ ( 1 − q k − 1 t ) . Then, from (3), we note that

From (31) and (32), we obtain the generating function of L q ( n , k ) as follows:

In this article, we consider the degenerate q-exponential functions, which are given by

When x = 1 , we have

When q = 1 , we note that

By (30), (31), and (34), we observe that

By comparing the coefficients of both sides of (37), we have

Now, we define the degenerate q-Bernoulli polynomials by

When x = 0 , B n , q ( λ ) = B n , q ( 0 ∣ λ ) are called the degenerate q-Bernoulli numbers.

When q = 1 , we note that B n , 1 ( x ∣ λ ) = B n , λ ( x ) are the degenerate Bernoulli polynomials.

By (34) and (39), we obtain the following properties.

The following are the first few values of B n , q ( λ ) :

We also consider the degenerate q-Euler polynomials given by

When x = 0 , E n , q ( λ ) = E n , q ( 0 ∣ λ ) , ( n ≥ 0 ) are called the degenerate q-Euler numbers.

When q = 1 , we note that E n , 1 ( x ∣ λ ) = E n , λ ( x ) are the degenerate Euler polynomials.

From (34) and (40), we obtain the following proposition.

The following are the first few values of E n , q ( λ ) :

For r ∈ N , we naturally consider the degenerate q-Bernoulli polynomials of order r as follows:

When x = 0 , B n , q ( r ) ( λ ) = B n , q ( r ) ( 0 ∣ λ ) are called the degenerate q-Bernoulli numbers of order r .

Then, we have

We also consider the degenerate q-Euler polynomials of order r given by

When x = 0 , E n , q ( r ) ( λ ) = E n , q ( r ) ( 0 ∣ λ ) are called the degenerate q-Euler numbers of order r .

From (34) and (43), we obtain

3 λ -q-Sheffer sequences and degenerate q-Sheffer sequences

In this section, we introduce a family of λ -linear functionals on the space of polynomials and prove some basic theorems and propositions on these functionals.

Throughout this section, λ is arbitrary but is a fixed nonzero real number.

Let f ( t ) = ∑ k = 0 ∞ a k [ k ] q ! t k ∈ F . Then each λ ∈ R gives rise to the linear functional ⟨ f ( t ) ∣ ⋅ ⟩ λ , q on P , called λ -q-linear functional given by f ( t ) , which is defined by

From (16) and (45), we note that

By (45), for f ( t ) ∈ F and p ( x ) in P , we have

and

From (12) and (13), we note that

Thus, we have