Hindawi Publishing Corporation Journal of Applied Mathematics Volume 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 Correspondence should be addressed to Sheng-liang Yang; [email protected] Received 5 June 2013; Accepted 25 July 2013 Academic Editor: Carla Roque Copyright © 2013 S.-l. Yang and S.-n. 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. 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 Let (, ) denote the coefficient of in −1().We call (, ) a signless Bessel number of the first kind and − The Bessel polynomials form a set of orthogonal polyno- (, ) = (−1) (, ) aBesselnumberofthefirst mials on the unit circle in the complex plane. They are kind. The Bessel number of the second kind (, ) is important in certain problems of mathematical physics; defined to be the number of set partitions of [] := for example, they arise in the study of electrical net- {1,2,3,...,} into blocks of size one or two. Han and works and when the wave equation is considered in spher- Seo [3] showed that the two kinds of Bessel numbers are ical coordinates. Many other important applications of related by inverse formulas, and both Bessel numbers of such polynomials may be found in Grosswald [1,pages the first kind and those of the second kind form log- 131–149]. The Bessel polynomials [1, 2]arethepoly- concave sequences. In [4], the Bessel matrices obtained nomial solutions of the second-order differential equa- from the Bessel numbers of the first kind (, ) and tions of the second kind (, ) are introduced, respectively. It is shown that these matrices can be represented by 2 () + (2 +) 2 () =(+1) () ,(0) =1. the exponential Riordan matrices. The relations between (1) the Bessel matrices, Catalan matrices, Stirling matrices, and Fibonacci matrices are also investigated. Recently, From this differential equation one can easily derive the [5] showed that Bessel numbers correspond to a special formula case of the generalized Stirling numbers. Bessel polyno- + mials have the following two determinant representations () = ∑ ( )( )! . (2) [6]: =0 2 2 Journal of Applied Mathematics 2 (1+) −1 0 ⋅⋅⋅ 0 −2 (2+) 2 (1+) −2 ⋅⋅⋅ 0 22 (3+) −2 (2+) 2 (1+) ⋅⋅⋅ 0 2 () = , . . d . −1 −2 −3 2(−) (+) 2(−) (−1+) 2(−) (−2+) ⋅⋅⋅ 2(1+) 2 − 1 −10 ⋅⋅⋅000 (3) ( ) 1 (2 −) 3 −1⋅⋅⋅000 01(2 −) 5 ⋅⋅⋅00 0 () = . . . d . 000⋅⋅⋅1−1 000⋅⋅⋅011 In Egecio˘ glu˘ [7], a ( + 1) × ( +1) Hankel determinant Lemma 1. Let =(,),≥0 = [(), ℎ()] be an exponential ̃ ̃ () =∑∞ () =∑∞ () is introduced by () = det[+()]0≤,≤,where Riordan matrix and let =0 , =0 be (ℎ()) =ℎ () (ℎ()) = () = ∑ ( /!),andaclosedformevaluationof two formal power series such that ,and =0 ()/() =(1/!)∑ !( + ) this determinant in terms of the Bessel polynomials is .Then +1, ≥−1 − −+1 ,, where defining −1 =0. Conversely, starting from the sequences given. Hence another determinant expression for the Bessel ∞ ∞ () =∑ () =∑ polynomials is given by (2) = () with () = =0 ,and =0 ,theinfinitearray (+1)/2 2 2 (,),≥0 defined by above relations is an exponential Riordan det[+()]0≤,≤,where =(−1) 2 ∏=1(! /(2)! ) matrix. and () = ∑=0( /(−)!). In this paper, using the production matrix of Bessel From this proposition, the production matrix of = matrix which is represented as exponential Riordan array, we [(), ℎ()] is =(,),≥0 with , = (!/!)(− +−+1) show that a variation of the generalized Bessel polynomial (−1 =0), where the numbers and are determined by ∞ ∞ sequence is of Sheffer type, and we give a determinant ()=0 =∑ =(())/(()), ()=0 =∑ = formula for the generalized Bessel polynomials. As a result, (()).Thesequence()≥0 and ()≥0 are called, respec- the Bessel polynomial is represented as determinant the tively, the -sequence and the -sequence of exponential entries of which involve Catalan numbers. Riordan array = [(), .ℎ()] Aswellknown,manypolynomialsequenceslikeLaguerre 2. Exponential Riordan Array polynomials, first and second kind Meixner polynomials, Poisson-Charlier polynomials, and Stirling polynomials form The concept of Riordan array was introduced by Shapiro Sheffer sequences. The Sheffer polynomials are specified here et al. [8] in 1991; then the concept is generalized to the by means of the following generating function (see [14, 15] exponential Riordan array by many authors [4, 9–13]. Let and the references cited therein): ∞ ∞ () = 1+ ∑=1 ( /!), ℎ() = ∑=1 ℎ( /!) with ℎ =0̸ ∞ 1 1 be two formal power series. An exponential Riordan ∑ () = (), =( ) (5) array is an infinite lower triangular array , ,≥0 =0 ! ( ()) whose th column has exponential generating function (1/!)()ℎ() . The array corresponding to the pair (), where () is a delta series and () an invertible series, ℎ() = [(), ℎ()] is denoted by .Thegrouplawis and () is the compositional inverse of ().Thesequence [(), ℎ()][(), ()] = [()(ℎ()), (ℎ())] then given by . () in (5) is called the Sheffer sequence for the pair From the definition it follows at once that the bivariate ((), ()).TheSheffersequence,() for (1, ()) is called = [(), ℎ()] (, ) = generating function of ,namely, theassociatedsequenceanditsatisfiesbinomialformula ∑ (/!) , , ,isgivenby (+)==0 ∑ ()−(),sowealsosaythatitis of binomial type. In [13],theauthorprovedthefollowing theorem ([13,Theorem2.4]). (,) =() exp ( ()) . (4) Lemma 2. Let [(), ()] be an exponential Riordan array = [(), ℎ()] Each exponential Riordan array is asso- with the -sequences ()≥0 and -sequences ()≥0.Let ciated a matrix called its production matrix [12], which is (())≥0 be the Sheffer polynomial sequence for ((), ()). −1 determined by = ,where is the matrix with its Then 0() =, 1 1()=−0,andforall≥0, +1() = top row removed. The production matrix of an exponential det(+1−+1),where+1 is the (+1)th principal submatrix Riordan array is characterized by the following lemma [10]. of the production matrix . Journal of Applied Mathematics 3 3. Bessel Polynomials and Bessel Matrices Proof. From Theorem 3,weobtain ≥≥0 ∞ ∞ For , the signless Bessel number of the − ∑ (,) = ∑ (2) ( ) first kind (, ) is defined to be the coefficient of in =0 ! =0 2 ! Bessel polynomial −1() with (0, 0). =1 The infinite lower triangular matrix with generic entry (, ) is called ∞ (2) = ∑ ( ) =((1−√1−4)/2). signless Bessel matrix of the first kind. From [4], the signless 2 ! Bessel matrix of the first kind can be represented as the =0 exponential Riordan array =[1,1−√1−2]. (10) Let () = −1(1/) for ≥1and 0() =.Then 1 Using (5), we obtain the desired result. the coefficients matrix of polynomial sequence {()}≥0 is the signless Bessel matrix of the first kind =[1,1−√1−2]. From the previous proof, the polynomial sequence Let () = (1/) for ≥0.Then() = (1/)+1(), (, ) has exponential generating function and () = (1/) ==0 ∑ (+1,+1).By[10], the infinite lower triangular matrix with generic entry ∞ ∑ (,) =((1−√1−4)/2), , = (+1,+1)is the exponential Riordan array = (11) =0 ! [1/√1−2,1−√1−2].WecallthismatrixtheBesselmatrix since it is the coefficients matrix of the Bessel polynomials. and () = (1/2, ).So{(, )}≥0 is a generalization of From (4), one has {()}≥0. Instead of working with (, ,,itissomewhateas- ) ∞ (1−√1−2) ier to work with the polynomials (,,) defined as ∑ () = , (6) ++−2 − ! (, , ) = ∑=0 ( )( )! ,whicharerelated =0 to the (, , ) by (, , ) = (, ,, ) ≥0.Itis ∞ 1 (1−√1−2) easy to check that (1/2, 2, ) = ().So{(, , ≥0)} is a ∑ () = . (7) {()}≥0 =0 ! √1−2 generalization of . Theorem 5. {(, , ≥0)} is a Sheffer polynomial sequence Henceweobtainthefollowingresults(seealso[14, 16]). in the variable . Theorem 3. { ()} −2 The polynomial sequence ≥0 is of bino- Proof. Let () = [(1/√1 − 4)((1 − √1−4)/2) ,(1− { ()} mial type, and the polynomial sequence ≥0 is a Sheffer √1−4)/2].By(5.72) of [17,page203],one sequence. has the general term of () which is (), = [/!](1/√1−4)((1 − √1−4)/2)+−2 = The generalized Bessel polynomial (, , ) was defined [/!] ∑∞ ( 2++−2 )() =( )(2−+−2 )(−)!− by Krall and Frink [2] as the polynomial solution of the =0 − . 2−+−2 differential equation Hence ∑=0 (), = ∑=0 ( )( − )( − − 2−+−2 − )! = ∑=0 ( − )( − )( − )! = 2 ++−2 − ++−2 () + () + () =(+−1) () , ∑=0( )( )! = ∑=0( )( )!(/) = (, , ) = (, ,. ) Therefore, the exponential gener- =0,̸ =0,−1,−2,....̸ ating function of the sequence {(, , ≥0)} is (8) ∞ 1 1−√1−4 −2 ∑ (, , ) = ( ) The explicit expression of the generalized Bessel polynomials ! √1−4 2 is =0 (12) ×((1−√1−4)/2). ++−2 (, ,) = ∑ ( )( )!( ) . (9) =0 Using (5), we obtain the desired result. In addition these polynomials satisfy the three-term recur- From Theorems 4 and 5, we get the following theorem. sion relation ( + − 1)(2 ++1 −2) () = [(2 + )(2 + Theorem 6. The sequences {(, )}≥0 and {(, , ≥0)} −2)(/)+−2](2+−1)() + (2−1 +) (), ≥2, satisfy the following convolution formulas: with