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Recurrence for probabilistic extension of Dowling polynomials

  • Yuankui Ma , Taekyun Kim EMAIL logo , Dae San Kim and Rongrong Xu
Published/Copyright: May 7, 2025
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Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

Spivey found a remarkable recurrence relation for Bell numbers, which was generalized to that for Bell polynomials by Gould-Quaintance. The aim of this article is to generalize their recurrence relation for Bell polynomials to that for the probabilistic Dowling polynomials associated with Y and also that for the probabilistic r -Dowling polynomials associated with Y . Here Y is a random variable whose moment generating function exists in a neighborhood of the origin.

Keywords: probabilistic Whitney numbers of the second kind; probabilistic Dowling polynomials; probabilistic r-Whitney numbers of the second; probabilistic r-Dowling polynomials
MSC 2010: 11B73; 11B83

1 Introduction

Assume that Y is a random variable whose moment generating function exists in a neighborhood of the origin (see (11)). We consider the probabilistic Whitney numbers of the second kind associated with Y , W m Y ( n , k ) (see (14)), as a probabilistic extension of the Whitney numbers of the second kind W m ( n , k ) (see (5), (7)). Here we note that the Whitney numbers of the second kind amount to the Stirling numbers of the second kind. Then, as a polynomial extension of W m Y ( n , k ) , we introduce the probabilistic Dowling polynomials associated with Y , D m Y ( n , x ) (see (16)), which is a probabilistic extension of the Dowling polynomials D m ( n , x ) (see (9)). The aim of this article is to generalize the Gould-Quaintance’s recurrence relation for Bell polynomials (see (3), (4)) to that for D m Y ( n , x ) , which is given by

(1) D m Y ( n + k , x ) = ∑ l = 0 n n l m n − l D m Y ( l , x ) ∑ i = 0 k ∑ j = 0 i k i x j m i − j j ! ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l i E [ Y 1 l 1 … Y j l j S j n − l ] .

We note here that (1) boils down to the following recurrence relation when Y = 1 :

D m ( n + k , x ) = ∑ l = 0 n ∑ j = 0 k n l m n − l j n − l W m ( k , j ) x j D m ( l , x ) , ( n , k ≥ 0 ) .

We also consider their probabilistic r -Whitney numbers of the second kind associated with Y , W m , r Y ( n , k ) (see (21)) and their polynomial extension, namely the probabilistic r -Dowling polynomials associated with Y , D m , r Y ( n , x ) (see (23)). Then we derive a recurrence relation that generalizes Gould-Quaintance’s for Bell polynomials (see (3), (4)). For the rest of this article, we recall the facts that are needed throughout the article.

It is known that the Bell polynomials are defined by

(2) ϕ n ( x ) = ∑ k = 0 n n k x k ( see [1–18] ) ,

with the Bell numbers given by ϕ n = ϕ n ( 1 ) , where n k are the Stirling numbers of the second kind.

Spivey found an interesting recurrence relation for ϕ n given in the following:

(3) ϕ l + n = ∑ k = 0 l ∑ i = 0 n k n − i n i l k ϕ i , ( l , n ≥ 0 ) ( see [19] ) .

In [20], Gould-Quaintance extended the recurrence relation for Bell numbers in (3) to that for Bell polynomials, which is given by

(4) ϕ l + n ( x ) = ∑ k = 0 n ∑ i = 0 n k n − i n i l k ϕ i ( x ) x i .

It is well known that the Whitney numbers of the second kind are defined by

(5) ( m x + 1 ) n = ∑ k = 0 n m k W m ( n , k ) ( x ) k , ( m ∈ N ) ( see [21,22,23] ) ,

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

For m = 1 , we have W 1 ( n , k ) = n + 1 k + 1 . For r ∈ N , the r -Whitney numbers of the second kind are defined by

(6) ( m x + r ) n = ∑ k = 0 n m k W m , r ( n , k ) ( x ) k , ( n ≥ 0 ) ( see [21,22,23] ) .

From (5) and (6), we note that

(7) e t 1 k ! e m t − 1 m k = ∑ n = k ∞ W m ( n , k ) t n n !

and

(8) e r t 1 k ! e m t − 1 m k = ∑ n = k ∞ W m , r ( n , k ) t n n ! ( see [21,22,23] ) .

The Dowling polynomials are defined by

(9) D m ( n , x ) = ∑ k = 0 n W m ( n , k ) x k , ( n ≥ 0 ) ( see [22] ) ,

and the r -Dowling polynomials are given by

(10) D m , r ( n , x ) = ∑ k = 0 n W m , r ( n , k ) x k , ( n ≥ 0 ) ( see [23] ) .

We assume that Y is a random variable satisfying the moment conditions

(11) E [ ∣ Y ∣ n ] < ∞ , n ∈ N ∪ { 0 } , lim n → ∞ ∣ t ∣ n E [ ∣ Y ∣ n ] n ! = 0 , ∣ t ∣ < r ,

for some r , where E stands for the mathematical expectation [24,25].

Let ( Y j ) j ≥ 1 be a sequence of mutually independent copies of the random variable Y , and let

(12) S 0 = 0 , S k = Y 1 + Y 2 + … + Y k , ( k ≥ 1 ) .

Recently, the probabilisitic Stirling numbers of the second kind associated with Y are given by

(13) n m Y = 1 m ! ∑ k = 0 m m k ( − 1 ) m − k E [ S k n ] , ( 0 ≤ m ≤ n ) ( see [24,25] ) .

2 Recurrence for probabilistic extension of Dowling polynomials

Let ( Y j ) j ≥ 1 be a sequence of mutually independent copies of the random variable Y , and let

S 0 = 0 , S k = Y 1 + Y 2 + … + Y k , ( k ≥ 1 ) .

In view of (7), we consider the probabilistic Whitney numbers of the second kind associated with Y given by

(14) 1 k ! E [ e m Y t ] − 1 m k e t = ∑ n = k ∞ W m Y ( n , k ) t n n ! , ( k ≥ 0 ) .

When Y = 1 , we have W m Y ( n , k ) = W m ( n , k ) .

By (14), we obtain

(15) ∑ n = k ∞ W m Y ( n , k ) t n n ! = 1 m k k ! ∑ j = 0 k k j ( − 1 ) k − j E [ e ( m ( Y 1 + Y 2 + … + Y j ) + 1 ) t ] = 1 m k k ! ∑ j = 0 k k j ( − 1 ) k − j E [ e ( m S j + 1 ) t ] = ∑ n = 0 ∞ 1 m k k ! ∑ j = 0 k k j ( − 1 ) k − j E [ ( m S j + 1 ) n ] t n n ! .

Therefore, by (15), we obtain the following theorem.

Theorem 1

For n ≥ k ≥ 0 , we have

W m Y ( n , k ) = 1 m k k ! ∑ j = 0 k k j ( − 1 ) k − j E [ ( m S j + 1 ) n ] .

In view of (9), we define the probabilistic Dowling polynomials associated with Y by

(16) D m Y ( n , x ) = ∑ k = 0 n W m Y ( n , k ) x k , ( n ≥ 0 ) .

From (16), we note that

(17) ∑ n = 0 ∞ D m Y ( n , x ) t n n ! = ∑ n = 0 ∞ ∑ k = 0 n W m Y ( n , k ) x k t n n ! = ∑ k = 0 ∞ x k e t 1 k ! E [ e m Y t ] − 1 m k = e t e x E [ e m Y t ] − 1 m .

Theorem 2

The generating function of Dowling polynomials is given by

e t e x E [ e m Y t ] − 1 m = ∑ n = 0 ∞ D m Y ( n , x ) t n n ! .

Using Taylor series, we note that

(18) f ( x + t ) = ∑ n = 0 ∞ f ( n ) ( x ) n ! t n = ∑ n = 0 ∞ t n D x n n ! f ( x ) = e t D x f ( x ) ,

where D x f ( x ) = d d x f ( x ) .

By (17) and (18), we obtain

(19) e z D t e t e x E [ e m Y t ] − 1 m = ∑ k = 0 ∞ z k k ! D t k ∑ n = 0 ∞ D m Y ( n , x ) t n n ! = ∑ k = 0 ∞ ∑ n = 0 ∞ D m Y ( n + k , x ) z k k ! t n n ! .

On the other hand, by (18), we obtain

(20) e z D t e t e x E [ e m Y t ] − 1 m = e t + z e x E [ e m Y ( t + z ) ] − 1 m = e t e x E [ e m Y t ] − 1 m e z e x E [ e m Y t ( e m Y z − 1 ) ] m = ∑ l = 0 ∞ D m Y ( l , x ) t l l ! ∑ k = 0 ∞ ∑ i = 0 k ∑ j = 0 i k i x j j ! m i m j ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l j E [ Y 1 l 1 … Y j l j e m S j t ] z k k ! = ∑ n = 0 ∞ ∑ k = 0 ∞ ∑ l = 0 n n l D m Y ( l , x ) ∑ i = 0 k ∑ j = 0 i k i x j m i − j j ! × ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l i E [ Y 1 l 1 … Y j l j S j n − l ] m n − l z k k ! t n n ! .

Therefore, by (19) and (20), we obtain the following theorem.

Theorem 3

For n , k ≥ 0 , we have

D m Y ( n + k , x ) = ∑ l = 0 n n l m n − l D m Y ( l , x ) ∑ i = 0 k ∑ j = 0 i k i x j m i − j j ! ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l i E [ Y 1 l 1 … Y j l j S j n − l ] .

In view of (8), we consider the probabilistic r-Whitney numbers of the second kind associated with Y given by

(21) 1 k ! E [ e m Y t ] − 1 m k e r t = ∑ n = k ∞ W m , r Y ( n , k ) t n n ! , ( k ≥ 0 ) .

When Y = 1 , we have W m , r Y ( n , k ) = W m , r ( n , k ) .

From (21), we note that

(22) ∑ n = k ∞ W m , r Y ( n , k ) t n n ! = 1 k ! m k ∑ j = 0 k k j ( − 1 ) k − j E [ e m ( Y 1 + … + Y j ) t ] e r t = 1 k ! m k ∑ j = 0 k k j ( − 1 ) k − j E [ e ( m S j + r ) t ] = ∑ n = 0 ∞ 1 k ! m k ∑ j = 0 k k j ( − 1 ) k − j E [ ( m S j + r ) n ] t n n ! .

Therefore, by (22), we obtain the following theorem.

Theorem 4

For n ≥ k ≥ 0 , we have

W m , r Y ( n , k ) = 1 k ! m k ∑ j = 0 k k j ( − 1 ) k − j E [ ( m S j + r ) n ] .

When Y = 1 , we have

W m , r ( n , k ) = 1 m k k ! ∑ j = 0 k k j ( − 1 ) k − j ( m j + r ) n .

In view of (16), we define the probabilistic r-Dowling polynomials associated with Y as

(23) D m , r Y ( n , x ) = ∑ k = 0 n W m , r Y ( n , k ) x k , ( n ≥ 0 ) .

When Y = 1 , we have D m , r Y ( n , x ) = D m , r ( n , x ) .

From (23), we have

(24) ∑ n = 0 ∞ D m , r Y ( n , x ) t n n ! = ∑ n = 0 ∞ ∑ k = 0 n W m , r Y ( n , k ) x k t n n ! = ∑ k = 0 ∞ x k ∑ n = k ∞ W m , r Y ( n , k ) t n n ! = e x E [ e m Y t ] − 1 m e r t .

Therefore, by (24), we obtain the following theorem.

Theorem 5

The generating function of probabilistic r-Dowling polynomials is given by

(25) e r t e x E [ e m Y t ] − 1 m = ∑ n = 0 ∞ D m , r Y ( n , x ) t n n ! .

By (25), we obtain

D m , r Y ( n , x ) = e − x m ∑ k = 0 ∞ x k k ! m k E [ ( m S k + r ) n ] , ( n ≥ 0 ) .

Theorem 6

For n ≥ 0 , we have

D m , r Y ( n , x ) = e − x m ∑ k = 0 ∞ x k k ! m k E [ ( m S k + r ) n ] .

Now, we observe that

(26) e z D t e r t e x E [ e m Y t ] − 1 m = ∑ k = 0 ∞ z k k ! D t k ∑ n = 0 ∞ D m , r Y ( n , x ) t n n ! = ∑ k = 0 ∞ ∑ n = 0 ∞ D m , r Y ( n + k , x ) t n n ! z k k ! .

On the other hand, by (18), we obtain

(27) e z D t e r t e x E [ e m Y t ] − 1 m = e r ( z + t ) e x E [ e m Y ( z + t ) ] − 1 m = e r z e x E [ e m Y z ] − 1 m e r t e x E [ e m Y z ( e m Y t − 1 ) ] − 1 m = ∑ l = 0 ∞ D m , r Y ( l , x ) z l l ! e r t ∑ j = 0 ∞ x j j ! E [ e m Y z ( e m Y t − 1 ) ] m j = ∑ l = 0 ∞ D m , r Y ( l , x ) z l l ! e r t ∑ j = 0 ∞ x j j ! m j E [ e m S j z ( e m Y 1 t − 1 ) … ( e m Y j t − 1 ) ] = ∑ l = 0 ∞ D m , r Y ( l , x ) z l l ! e r t ∑ i = 0 ∞ ∑ j = 0 i m i − j x j j ! ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l j × E [ Y 1 l 1 Y 2 l 2 … Y j l j e m S j z ] t i i ! = ∑ l = 0 ∞ D m , r Y ( l , x ) z l l ! ∑ k = 0 ∞ ∑ i = 0 k k i r k − i ∑ j = 0 i m i − j x j j ! ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l j × E [ Y 1 l 1 Y 2 l 2 … Y j l j e m S j z ] t k k ! = ∑ n = 0 ∞ ∑ k = 0 ∞ ∑ l = 0 n n l D m , r Y ( l , x ) ∑ i = 0 k k i r k − i ∑ j = 0 i m i − j j ! x j ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l j × E [ Y 1 l 1 Y 2 l 2 … Y j l j ( m S j ) n − l ] z n n ! t k k ! .

Thus, by (26) and (27), we obtain

(28) D m , r Y ( n + k , x ) = ∑ l = 0 n n l m n − l D m , r Y ( l , x ) ∑ i = 0 k k i r k − i ∑ j = 0 i m i − j j ! x j × ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l j E ∏ m = 1 j Y m l m S j n − l .

When Y = 1 , we have

D m , r ( n + k , x ) = ∑ l = 0 n ∑ j = 0 k W m , r ( k , j ) x j n l D m , r ( l , x ) m n − l j n − l , ( n , k ≥ 0 ) .

For this, one has to observe that

W m , r ( k , j ) = 1 m j ∑ i = j k k i r k − i m i 1 j ! ∑ l 1 + … + l j = i l 1 , … , l j ≥ 1 i l 1 , … , l j = 1 m j ∑ i = j k k i r k − i m i i j ,

where the last identity follows from (8).

Theorem 7

For n , k ≥ 0 , we have

D m , r Y ( n + k , x ) = ∑ l = 0 n n l m n − l D m , r Y ( l , x ) ∑ i = 0 k k i r k − i ∑ j = 0 i m i − j j ! x j ∑ l 1 + … + l j = i l 1 , l 2 , … , l j ≥ 1 i l 1 , … , l j E ∏ m = 1 j Y m l m S j n − l .

3 Conclusion

Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. We derived recurrence relations for the probabilistic Dowling polynomials associated with Y , D m Y ( n , x ) and the probabilistic r -Dowling polynomials associated with Y , D m , r Y ( n , x ) , which generalized the recurrence relation for Bell polynomials due to Gould-Quaintance. In detail, an expression for W m Y ( n , k ) was derived in Theorem 1. We obtained the generating function and a recurrence relation of D m Y ( n , x ) , respectively, in Theorem 2 and 3. An expression for W m , r Y ( n , k ) was given in Theorem 4. We found the generating function and an expression for D m , r Y ( n , x ) , respectively, in Theorems 5 and 6. Finally, we derived a recurrence relation for D m , r Y ( n , x ) in Theorem 7.

As one of our future projects, we would like to continue to study probabilistic extensions of many special polynomials and numbers and to find their applications to physics, science, and engineering as well as to mathematics.

Acknowledgements

We thank the referees for their helpful comments and suggestions. We also extend our gratitude to the Jangjeon Institute for Mathematical Sciences for their support of this research.

  1. Funding information: This research was funded by the National Natural Science Foundation of China (No. 12271320), Key Research and Development Program of Shaanxi (No. 2023-ZDLGY-02).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: No data were used to support this study.

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Received: 2024-03-08
Revised: 2025-04-04
Accepted: 2025-04-05
Published Online: 2025-05-07

© 2025 the author(s), published by De Gruyter

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

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  68. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
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  75. Green's graphs of a semigroup
  76. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  77. Infinitely many solutions for a class of Kirchhoff-type equations
  78. On an uncertainty principle for small index subgroups of finite fields
https://doi.org/10.1515/math-2025-0150
Keywords for this article
probabilistic Whitney numbers of the second kind; probabilistic Dowling polynomials; probabilistic r-Whitney numbers of the second; probabilistic r-Dowling polynomials
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