Journal of Applied Mathematics

Volume 2013 (2013), Article ID 242815, 6 pages

http://dx.doi.org/10.1155/2013/242815

Research Article

A Determinant Expression for the Generalized Bessel Polynomials

Sheng-liang Yang and Sai-nan Zheng

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 5 June 2013; Accepted 25 July 2013

Academic Editor: Carla Roque

Copyright © 2013 Sheng-liang Yang and Sai-nan Zheng. 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.

Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.

1. Introduction

The Bessel polynomials form a set of orthogonal polynomials on the unit circle in the complex plane. They are important in certain problems of mathematical physics; for example, they arise in the study of electrical networks and when the wave equation is considered in spherical coordinates. Many other important applications of such polynomials may be found in Grosswald [1, pages 131–149]. The Bessel polynomials [1, 2] are the polynomial solutions of the second-order differential equations

From this differential equation one can easily derive the formula

Let denote the coefficient of in . We call a signless Bessel number of the first kind and a Bessel number of the first kind. The Bessel number of the second kind is defined to be the number of set partitions of into blocks of size one or two. Han and Seo [3] showed that the two kinds of Bessel numbers are related by inverse formulas, and both Bessel numbers of the first kind and those of the second kind form log-concave sequences. In [4], the Bessel matrices obtained from the Bessel numbers of the first kind and of the second kind are introduced, respectively. It is shown that these matrices can be represented by the exponential Riordan matrices. The relations between the Bessel matrices, Catalan matrices, Stirling matrices, and Fibonacci matrices are also investigated. Recently, [5] showed that Bessel numbers correspond to a special case of the generalized Stirling numbers. Bessel polynomials have the following two determinant representations [6]:

In Eğecioğlu [7], a Hankel determinant is introduced by , where , and a closed form evaluation of this determinant in terms of the Bessel polynomials is given. Hence another determinant expression for the Bessel polynomials is given by with , where and .

In this paper, using the production matrix of Bessel matrix which is represented as exponential Riordan array, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we give a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.

2. Exponential Riordan Array

The concept of Riordan array was introduced by Shapiro et al. [8] in 1991; then the concept is generalized to the exponential Riordan array by many authors [4, 9–13]. Let , with be two formal power series. An exponential Riordan array is an infinite lower triangular array whose th column has exponential generating function . The array corresponding to the pair , is denoted by . The group law is then given by . From the definition it follows at once that the bivariate generating function of , namely, , is given by

Each exponential Riordan array is associated a matrix called its production matrix [12], which is determined by , where is the matrix with its top row removed. The production matrix of an exponential Riordan array is characterized by the following lemma [10].

Lemma 1. Let be an exponential Riordan matrix and let , be two formal power series such that , and . Then , where defining . Conversely, starting from the sequences , and , the infinite array defined by above relations is an exponential Riordan matrix.

From this proposition, the production matrix of is with (), where the numbers and are determined by , . The sequence and are called, respectively, the -sequence and the -sequence of exponential Riordan array .

As well known, many polynomial sequences like Laguerre polynomials, first and second kind Meixner polynomials, Poisson-Charlier polynomials, and Stirling polynomials form Sheffer sequences. The Sheffer polynomials are specified here by means of the following generating function (see [14, 15] and the references cited therein):

where is a delta series and an invertible series, and is the compositional inverse of . The sequence in (5) is called the Sheffer sequence for the pair . The Sheffer sequence, for is called the associated sequence and it satisfies binomial formula , so we also say that it is of binomial type. In [13], the author proved the following theorem ([13, Theorem 2.4]).

Lemma 2. Let be an exponential Riordan array with the -sequences and -sequences . Let be the Sheffer polynomial sequence for . Then , , and for all , , where is the th principal submatrix of the production matrix .

3. Bessel Polynomials and Bessel Matrices

For , the signless Bessel number of the first kind is defined to be the coefficient of in Bessel polynomial with . The infinite lower triangular matrix with generic entry is called signless Bessel matrix of the first kind. From [4], the signless Bessel matrix of the first kind can be represented as the exponential Riordan array .

Let for and . Then the coefficients matrix of polynomial sequence is the signless Bessel matrix of the first kind . Let for . Then , and . By [10], the infinite lower triangular matrix with generic entry is the exponential Riordan array . We call this matrix the Bessel matrix since it is the coefficients matrix of the Bessel polynomials. From (4), one has

Hence we obtain the following results (see also [14, 16]).

Theorem 3. The polynomial sequence is of binomial type, and the polynomial sequence is a Sheffer sequence.

The generalized Bessel polynomial was defined by Krall and Frink [2] as the polynomial solution of the differential equation

The explicit expression of the generalized Bessel polynomials is

In addition these polynomials satisfy the three-term recursion relation, , with initial conditions and .

We introduce a two-variable polynomial sequence by . Then .

Theorem 4. is a sequence of polynomials of binomial type in the variable .

Proof. From Theorem 3, we obtain

Using (5), we obtain the desired result.

From the previous proof, the polynomial sequence has exponential generating function

and . So is a generalization of .

Instead of working with , it is somewhat easier to work with the polynomials defined as , which are related to the by , . It is easy to check that . So is a generalization of .

Theorem 5. is a Sheffer polynomial sequence in the variable .

Proof. Let . By of [17, page 203], one has the general term of which is             . Hence   =                      . Therefore, the exponential generating function of the sequence is

Using (5), we obtain the desired result.

From Theorems 4 and 5, we get the following theorem.

Theorem 6. The sequences and satisfy the following convolution formulas:

Note that the coefficients matrix of the polynomial sequence is , which is a generalization of the signless Bessel matrix of the first kind while the coefficients matrix of the polynomial sequence is , which is a generalization of the Bessel matrix .

4. Determinant Formulae for Bessel Polynomials

The Catalan numbers, , may be defined by the ordinary generating function . The formulae (11) and (12) imply the connection between the generalized Bessel polynomials and Catalan numbers. In the following we will give some determinant expressions for Bessel polynomials, in which the entries involve Catalan numbers.

Define to be the infinite Hessenberg matrix with general entry

The first rows of are

Theorem 7. For , the polynomial admits the following determinant representation , that is,

where is the th principal submatrix of Hessenberg matrix defined by (14).

Proof. Let and ; then , . Therefore, from (12), is a Sheffer polynomial sequence for the pair .

The generating functions for the -sequence and -sequence of are   =    =    =  ,   =    =    =  . By Lemma 1, the production matrix of is exactly the Hessenberg matrix defined by (14). Hence, by Lemma 2, we obtain the desired result.

Corollary 8. For , one has . Particularly,

Corollary 9. For , we have

Examples. One has

One has

Consider the following:

Consider the following:

One has

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11261032) and the Natural Science Foundation of Gansu Province (Grant no. 1010RJZA049).

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