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.