Recurrence for probabilistic extension of Dowling polynomials

Yuankui Ma; Taekyun Kim; Dae San Kim; Rongrong Xu

doi:10.1515/math-2025-0150

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Recurrence for probabilistic extension of Dowling polynomials

Yuankui Ma , Taekyun Kim , Dae San Kim und Rongrong Xu

Veröffentlicht/Copyright: 7. Mai 2025

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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.

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).
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.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data were used to support this study.

References

[1] M. Abbas and S. Bouroubi, On new identities for Bell’s polynomials, Discrete Math. 293 (2005), no. 1–3, 5–10, DOI: https://doi.org/10.1016/j.disc.2004.08.023. 10.1016/j.disc.2004.08.023Suche in Google Scholar

[2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992. Suche in Google Scholar

[3] H. Belbachir and I. E. Bousbaa Translated Whitney and r-Whitney numbers: a combinatorial approach, J. Integer Seq. 16 (2013), no. 8, 13.8.6. Suche in Google Scholar

[4] K. Boubellouta, A. Boussayoud, S. Araci, and M. Kerada, Some theorems on generating functions and their applications, Adv. Stud. Contemp. Math. (Kyungshang) 30 (2020), no. 3, 307–324.Suche in Google Scholar

[5] A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), no. 3, 241–259, DOI: https://doi.org/10.1016/0012-365X(84)90161-4. 10.1016/0012-365X(84)90161-4Suche in Google Scholar

[6] L. Carlitz, Some remarks on the Bell numbers, Fibonacci Quart. 18 (1980), no. 1, 66–73, DOI: https://doi.org/10.1080/00150517.1980.12430191. 10.1080/00150517.1980.12430191Suche in Google Scholar

[7] L. Carlitz, Weighted Stirling numbers of the first and second kind II, Fibonacci Quart. 18 (1980), no. 3, 242–257, DOI: https://doi.org/10.1080/00150517.1980.12430154. 10.1080/00150517.1980.12430154Suche in Google Scholar

[8] L. Carlitz, Weighted Stirling numbers of the first and second kind I, Fibonacci Quart. 18 (1980), no. 2, 147–162, DOI: https://doi.org/10.1080/00150517.1980.12430168. 10.1080/00150517.1980.12430168Suche in Google Scholar

[9] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., Dordrecht, 1974. 10.1007/978-94-010-2196-8Suche in Google Scholar

[10] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd edition, Addison-Wesley Publishing Company, Reading, MA, 1994. Suche in Google Scholar

[11] N. Kilar and Y. Simsek, Combinatorial sums involving Fubini type numbers and other special numbers and polynomials: approach trigonometric functions and p-adic integrals, Adv. Stud. Contemp. Math. (Kyungshang) 31 (2021), no. 1, 75–87.Suche in Google Scholar

[12] J. Riordan, An Introduction to Combinatorial Analysis, Dover Publications, Inc., Mineola, NY, 2002. Suche in Google Scholar

[13] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. Suche in Google Scholar

[14] S. Roman, The Umbral Calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, publishers], New York, 1984, DOI: https://doi.org/10.1007/978-1-4757-2178-2_17. 10.1007/978-1-4757-2178-2_17Suche in Google Scholar

[15] S. M. Ross, Introduction to Probability Models, 13th edition, Academic Press, London, 2024. 10.1016/B978-0-44-318761-2.00006-3Suche in Google Scholar

[16] S. Soni, P. Vellaisamy, and A. K. Pathak, A probabilistic generalization of the Bell polynomials, J. Anal. 32 (2024), 711–732, DOI: https://doi.org/10.1007/s41478-023-00642-y. 10.1007/s41478-023-00642-ySuche in Google Scholar

[17] B. Q. Ta, Probabilistic approach to Appell polynomials, Expo. Math. 33 (2015), no. 3, 269–294, DOI: https://doi.org/10.1016/j.exmath.2014.07.003. 10.1016/j.exmath.2014.07.003Suche in Google Scholar

[18] Y. Zheng and N. N. Li, Bivariate extension of Bell polynomials, J. Integer Seq. 22 (2019), no. 8, 19.8.8. Suche in Google Scholar

[19] M. Z. Spivey, A generalized recurrence for Bell numbers, J. Integer Seq. 11 (2008), no. 2, 08.2.5. Suche in Google Scholar

[20] H. W. Gould and J. Quaintance, Implications of Spiveyas Bell number formula, J. Integer Seq. 11 (2008), no. 3, 08.3.7. Suche in Google Scholar

[21] D. S. Kim and T. Kim, Moment representations of fully degenerate Bernoulli and degenerate Euler polynomials, Russ. J. Math. Phys. 31 (20224), no. 4, 682–690, DOI: https://doi.org/10.1134/S1061920824040071. 10.1134/S1061920824040071Suche in Google Scholar

[22] T. Kim and D. S. Kim, Degenerate Whitney numbers of first and second kind of Dowling lattices, Russ. J. Math. Phys. 29 (2022), no. 3, 358–377, DOI: https://doi.org/10.1134/S1061920822030050. 10.1134/S1061920822030050Suche in Google Scholar

[23] T. Kim and D. S. Kim, Combinatorial identities involving degenerate harmonic and hyperharmonic numbers, Adv. Appl. Math. 148 (2023), 102535, DOI: https://doi.org/10.1016/j.aam.2023.102535. 10.1016/j.aam.2023.102535Suche in Google Scholar

[24] J. A. Adell, Probabilistic Stirling numbers of the second kind and applications, J. Theoret. Probab. 35 (2022), no. 1, 636–652, DOI: https://doi.org/10.1007/s10959-020-01050-9. 10.1007/s10959-020-01050-9Suche in Google Scholar

[25] T. Kim and D. S. Kim, Probabilistic degenerate Bell polynomials associated with random variables, Russ. J. Math. Phys. 30 (2023), no. 4, 528–542, DOI: https://doi.org/10.1134/S106192082304009X. 10.1134/S106192082304009XSuche in Google Scholar

Received: 2024-03-08

Revised: 2025-04-04

Accepted: 2025-04-05

Published Online: 2025-05-07

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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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