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Identities associated with Milne–Thomson type polynomials and special numbers
- Yilmaz Simsek^{1} and
- Nenad Cakic^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1679-x
© The Author(s) 2018
Received: 27 February 2018
Accepted: 3 April 2018
Published: 13 April 2018
Abstract
The purpose of this paper is to give identities and relations including the Milne–Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By using fermionic and bosonic p-adic integrals, we derive some new relations and formulas related to these numbers and polynomials, and also the combinatorial sums.
Keywords
- Generating function
- Functional equation
- Bernoulli numbers and polynomials
- Euler numbers and polynomials
- Stirling numbers
- Array polynomials
- Milne–Thomson polynomials
- Hermite polynomials
- Central factorial numbers
- Cauchy numbers
- Special functions
- p-adic integral
MSC
- 05A15
- 05A10
- 11B68
- 11B83
1 Introduction
Recently, many authors have studied special numbers and polynomials with their generating functions. Because these special numbers and polynomials including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Milne–Thomson numbers and polynomials, the Hermite numbers and polynomials, Central factorial numbers, Cauchy numbers, and the others have many applications not only in mathematics, but also in other related areas. It is well-known that there are also many combinatorial interpretations of these special numbers especially, the Stirling numbers and the central factorial numbers in partition theory, in set theory, in probability theory and in other sciences. For combinatorial interpretations of these special numbers and polynomials with their generating functions see for details [1–31], and the references therein.
In this paper the following notation is used.
In order to prove identities, relations, formulas, and combinatorial sums related to the special numbers and polynomials of this paper, we need the following generating functions for these special numbers and polynomials including some basic properties of them.
1.1 p-adic integral
Remark 1
Remark 2
We now summarize the results of this paper as follows:
In Sect. 2, by using generating functions and their functional equations, we give some identities including the three-variable polynomials \(y_{6} ( n;x,y,z;a,b,v ) \), the Hermite polynomials, the array polynomials and the Stirling numbers of the second kind.
In Sect. 3, by using p-adic integrals, we give some identities, combinatorial sums and relations related to the three-variable polynomials \(y_{6} ( n;x,y,z;a,b,v ) \), the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind) and other special numbers such as the Daehee numbers and the Changhee numbers.
In Sect. 4, by using generating functions associated with trigonometric functions and the central factorial numbers of the second kind, we derive identities related to the central factorial numbers of the second kind, the array polynomials, and combinatorial sum.
2 Identities related to the Hermite polynomials, array polynomials and Stirling numbers of the second kind: generating functions and their functional equations approach
In this section, by applying generating functions and their functional approach, we derive some identities including the three-variable polynomials \(y_{6} ( n;x,y,z;a,b,v ) \), the Hermite polynomials, the array polynomials and the Stirling numbers of the second kind.
Theorem 1
Proof
Theorem 2
Proof
Combining (18) and (19), we arrive at the following theorem.
Theorem 3
3 Identities and relations related to Stirling numbers and other special numbers: p-adic integral approach
In this section, by applying p-adic integrals approach, we derive some identities, combinatorial sums and relations related to the three-variable polynomials \(y_{6} ( n;x,y,z;a,b,v ) \), the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind) and other special numbers such as the Daehee numbers and the Changhee numbers.
Theorem 4
Proof
It is time to give integral formulas for the three-variable polynomials \(y_{6} ( n;x,y,z;a,b,v ) \).
Theorem 5
Theorem 6
Corollary 1
4 Identities including the central factorial numbers of the second kind and array polynomials
In this section, by using special infinite series including trigonometric functions and the central factorial numbers of the second kind, we derive identities associated with the central factorial numbers of the second kind and the array polynomials.
Theorem 7
By combining (29) and (30), we also get following corollary.
Corollary 2
With the help of Eq. (31), we also obtain the following combinatorial sum.
Corollary 3
5 Conclusion
This paper contains many kind of identities and relations related to the Milne–Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By applying not only p-adic integral, but also the Riemann integral methods, many identities relations and formulas related to the aforementioned numbers and polynomials, and also the combinatorial sums are given. By using the orthogonality relation of the Stirling numbers, explicit formulas for the Bernoulli numbers and the Euler numbers are provided.
The results of this paper have potential applicability to physics, engineering and other related fields, especially branches of mathematics.
Declarations
Acknowledgements
We would like to thank the referees for their valuable comments. The first author was supported by the Scientific Research Project Administration of Akdeniz University.
Authors’ contributions
All the authors participated in every phase of the research conducted for this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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