A Determinant Expression for the Generalized Bessel Polynomials

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


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  2    () + (2 + 2)    () =  ( + 1)   () ,   (0) = 1. ( From this differential equation one can easily derive the formula Let (, ) denote the coefficient of  − in  −1 ().We call (, ) a signless Bessel number of the first kind and (, ) = (−1) − (, ) a Bessel number of the first kind.The Bessel number of the second kind (, ) is defined to be the number of set partitions of [] := {1, 2, 3, . . ., } 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 logconcave 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 Egecioglu [7], a ( + 1) × ( + 1) Hankel determinant H () is introduced by H () = det[ + ()] 0≤,≤ , 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 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.

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][10][11][12][13].Let = 0 be two formal power series.An exponential Riordan array is an infinite lower triangular array  = ( , ) ,≥0 whose th column has exponential generating function From the definition it follows at once that the bivariate generating function of  = [(), ℎ()], namely,   (, ) = ∑ ,  ,   (  /!), is given by Each exponential Riordan array  = [(), ℎ()] is associated a matrix  called its production matrix [12], which is determined by  =  −1 , where  is the matrix  with its top row removed.The production matrix of an exponential Riordan array is characterized by the following lemma [10].
where defining  −1 = 0. Conversely, starting from the sequences () = ∑ ∞ =0     , and () = ∑ ∞ =0     , the infinite array ( , ) ,≥0 defined by above relations is an exponential Riordan matrix.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):

Bessel Polynomials and Bessel Matrices
For  ≥  ≥ 0, the signless Bessel number of the first kind (, ) is defined to be the coefficient of  − in Bessel polynomial  −1 () with (0, 0) = 1.The infinite lower triangular matrix  with generic entry (, ) is called signless Bessel matrix of the first kind.From [4] 4), one has Hence we obtain the following results (see also [14,16]).
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  0 () = 1 and  1 () = 1 + (/).
Theorem 4. {  (, )} ≥0 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 Theorems 4 and 5, we get the following theorem.( Note that the coefficients matrix of the polynomial sequence which is a generalization of the signless Bessel matrix of the first kind  = [1,1 − √ 1 − 2] while the coefficients matrix of the polynomial sequence