- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2014 (2014), Article ID 536325, 4 pages

http://dx.doi.org/10.1155/2014/536325

## On Certain Matrices of Bernoulli Numbers

^{1}The Institute of Applied Mathematics, College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China^{2}Department of Financial Engineering and Actuarial Mathematics, Soochow University, 56 Kueiyang Street, Sec. 1, Taipei 100, Taiwan

Received 3 May 2014; Revised 16 July 2014; Accepted 24 July 2014; Published 5 August 2014

Academic Editor: Guo-Cheng Wu

Copyright © 2014 Ruiming Zhang and Li-Chen Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work we compute the determinant and inverse matrices for a certain symmetric matrix of Rayleigh sums. As a special case we also obtain the determinants and inverses for the matrices of the Bernoulli numbers and related numbers.

#### 1. Introduction

The sequence of Bernoulli numbers is one of the most important sequences in mathematics. It has deep connections to number theory, for instance, the Bernoulli numbers are used to express the values of , where is the Riemann zeta function and is a positive integer [1, 2]. The Bernoulli numbers are also very important in analysis, for example, they appear in the Euler-Maclaurin formula [1], which is very important in mathematics and physics. The Bernoulli numbers are also very important in asymptotics of -special functions; for example, in [3] we proved a complete asymptotic expansion of -Gamma function on the complex plane in terms of Bernoulli polynomials and Bernoulli polynomials. The applications of Bernoulli numbers in applied mathematics are just too many to list all of them; just to name a few, for example, see [4–6]. The Rayleigh sums generalize and it is known that is a rational multiple of [7]. In this work we first derive the inverse and determinant of a certain symmetric matrix defined by and then specialize the result to the matrices defined by Bernoulli numbers and related numbers .

But we have to emphasize that the present work demonstrated a method to compute inverses of certain Hankel matrices, not just determinants. In fact there are many known methods to compute determinants; for example, see [1, 8–11].

#### 2. Preliminaries

For the Bessel function of first kind is defined by [1, 7, 11, 12]: where As a special case we have It is known that the even entire function has infinitely many zeros, all of which are real. Let be all its positive zeros; then the Rayleigh sum is defined by [7] Clearly [1], where the Bernoulli numbers are defined by [1, 2, 12] The related numbers are defined by [2, 13] for and ; it is known that

#### 3. Main Results

Theorem 1. *Given a nonnegative integer , one has
**
for .*

Corollary 2. *For any nonnegative integer , one has
**
or, equivalently,
*

#### 4. Proofs

Given a probability measure on such that for all , we define the inner product for square integrable functions and by For each , let with for where is a sequence of polynomials with such that, for each , are linearly independent. Then there is a unique orthonormal system [1, 10, 11]: with positive leading coefficient in . Clearly we have for some real numbers for and for .

Lemma 3. *For each nonnegative integer , let and . Then
*

*Proof. *From (15) and it is clear that
For each , since both and are a basis for the same set of polynomials, must be invertible for each . We denote ; then for . Clearly, for . Thus,
for , which is
and hence .

##### 4.1. Proof of Theorem 1

The normalized even order Lommel polynomials are defined by [11] for and . They satisfy the orthogonal relation For , it is clear that the th moment with respect to the measure of orthogonality is Let for ; then By Lemma 3, the matrix has determinant and its inverse has elements

##### 4.2. Proof of Corollary 2

From (24), (25), and (6), we get They are simplified to By (9) we get which are simplified to (12) and (13), respectively.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The first and corresponding author of this work, Ruiming Zhang, is partially supported by the National Natural Science Foundation of China, Grant no. 11371294. He also thanks Professors Jyh-Hao Lee, Derchyi Wu, for their hospitalities during his visits to Institute of Mathematics, Academia Sinica, Taipei.

#### References

- G. E. Andrews, R. Askey, and R. Roy,
*Special Functions*, vol. 71 of*Encyclopedia of Mathematics and its Applications*, Cambridge University Press, Cambridge, UK, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - Wikipedia, http://en.wikipedia.org/wiki/Bernoulli_number.
- R. Zhang, “On asymptotics of the {$q$}-exponential and {$q$}-gamma functions,”
*Journal of Mathematical Analysis and Applications*, vol. 411, no. 2, pp. 522–529, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - E. Tohidi and F. Toutounian, “Convergence analysis of Bernoulli matrix approach for one-dimensional matrix hyperbolic equations of the first order,”
*Computers & Mathematics with Applications*, vol. 68, no. 1-2, pp. 1–12, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - F. Toutounian, E. Tohidi, and S. Shateyi, “A collocation method based on the Bernoulli operational matrix for solving high-order linear complex differential equations in a rectangular domain,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 823098, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - F. Toutounian and E. Tohidi, “A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis,”
*Applied Mathematics and Computation*, vol. 223, pp. 298–310, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - G. N. Watson,
*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, New York, NY, USA, 2nd edition, 1944. View at MathSciNet - W. A. Al-Salam and L. Carlitz, “Some determinants of Bernoulli, Euler and related numbers,”
*Portugaliae Mathematica*, vol. 18, pp. 91–99, 1959. View at Zentralblatt MATH · View at MathSciNet - C. Krattenthaler, “Advanced determinant calculus,”
*Séminaire Lotharingien de Combinatoire*, vol. 42, article B42q, 67 pages, 1999. View at MathSciNet - T. S. Chihara,
*An Introduction to Orthogonal Polynomials*, Gordon and Breach, New York, NY, USA, 1978. View at MathSciNet - M. E. H. Ismail,
*Continuous and Discrete Orthogonal Polynomials*, Cambridge University Press, Cambridge, UK, 2005. - R. Zhang, “Sums of zeros for certain special functions,”
*Integral Transforms and Special Functions*, vol. 21, no. 5-6, pp. 351–365, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - N. D. Elkies, “On the sums ${\sum}_{k=-\infty}^{\infty}{(4k+1)}^{-n}$,”
*The American Mathematical Monthly*, vol. 110, no. 7, pp. 561–573, 2003. View at Publisher · View at Google Scholar · View at MathSciNet