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.

𝑛−𝑘 1. Introduction Let 𝑎(𝑛, 𝑘) denote the coefficient of 𝑧 in 𝑦𝑛−1(𝑧).We call 𝑎(𝑛, 𝑘) a signless Bessel number of the first kind and 𝑛−𝑘 The Bessel polynomials form a set of orthogonal polyno- 𝑏(𝑛, 𝑘) = (−1) 𝑎(𝑛, 𝑘) aBesselnumberofthefirst mials on the unit circle in the complex plane. They are kind. The Bessel number of the second kind 𝐵(𝑛, 𝑘) is important in certain problems of mathematical physics; defined to be the number of set partitions of [𝑛] := for example, they arise in the study of electrical net- {1,2,3,...,𝑛} into 𝑘 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 polynomials1 [ , 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 𝑏(𝑛, 𝑘) and tions of the second kind 𝐵(𝑛, 𝑘) are introduced, respectively. It is shown that these matrices can be represented by 𝑧2𝑦󸀠󸀠 (𝑧) + (2𝑧 + 2) 𝑦󸀠 (𝑧) =𝑛(𝑛+1) 𝑦 (𝑧) ,𝑦(0) =1. 𝑛 𝑛 𝑛 𝑛 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- 𝑛 𝑛 𝑛+𝑘 𝑧𝑘 mials have the following two determinant representations 𝑦 (𝑧) = ∑ ( )( )𝑘! . 𝑛 𝑘 𝑘 𝑘 (2) [6]: 𝑘=0 2 2 Journal of Applied Mathematics

󵄨 󵄨 󵄨 2 (1+𝑛𝑧) −1 0 ⋅⋅⋅ 0 󵄨 󵄨 󵄨 󵄨 −2𝑧 (2+𝑛𝑧) 2 (1+𝑛𝑧) −2 ⋅⋅⋅ 0 󵄨 󵄨 󵄨 𝑛 󵄨 2𝑧2 (3+𝑛𝑧) −2𝑧 (2+𝑛𝑧) 2 (1+𝑛𝑧) ⋅⋅⋅ 0 󵄨 2 𝑦 (𝑧) = 󵄨 󵄨 , 𝑛 󵄨 󵄨 󵄨 . . . . 󵄨 󵄨 . . . d . 󵄨 󵄨 𝑛−1 𝑛−2 𝑛−3 󵄨 󵄨2(−𝑧) (𝑛+𝑛𝑧) 2(−𝑧) (𝑛−1+𝑛𝑧) 2(−𝑧) (𝑛−2+𝑛𝑧) ⋅⋅⋅ 2(1+𝑛𝑧)󵄨 󵄨 2𝑛 − 1 𝑧−10 ⋅⋅⋅000󵄨 (3) 󵄨( ) 󵄨 󵄨 󵄨 󵄨 1 (2𝑛 − 3) 𝑧−1⋅⋅⋅000󵄨 󵄨 󵄨 󵄨 01(2𝑛 − 5) 𝑧⋅⋅⋅00󵄨 0 𝑦 (𝑧) = 󵄨 󵄨 . 𝑛 󵄨 ...... 󵄨 󵄨 . . . d . . . 󵄨 󵄨 󵄨 󵄨 000⋅⋅⋅1𝑧−1󵄨 󵄨 󵄨 󵄨 000⋅⋅⋅011󵄨

In Egecio˘ glu˘ [7], a (𝑛 + 1) × (𝑛 +1) Hankel determinant Lemma 1. Let 𝐴=(𝑎𝑛,𝑘)𝑛,𝑘≥0 = [𝑔(𝑡), ℎ(𝑡)] be an exponential ̃ ̃ 𝑐(𝑡) =∑∞ 𝑐 𝑡𝑛 𝑟(𝑡) =∑∞ 𝑟 𝑡𝑛 𝐻𝑛(𝑧) is introduced by 𝐻𝑛(𝑧) = det[𝑒𝑖+𝑗(𝑧)]0≤𝑖,𝑗≤𝑛,where Riordan matrix and let 𝑛=0 𝑛 , 𝑛=0 𝑛 be 󸀠 𝑘 𝑚 𝑟(ℎ(𝑡)) =ℎ (𝑡) 𝑐(ℎ(𝑡)) = 𝑒𝑘(𝑧) = ∑ (𝑧 /𝑚!),andaclosedformevaluationof two formal power series such that ,and 𝑚=0 𝑔󸀠(𝑡)/𝑔(𝑡) 𝑎 =(1/𝑘!)∑ 𝑖!(𝑐 +𝑘𝑟 )𝑎 this determinant in terms of the Bessel polynomials is .Then 𝑛+1,𝑘 𝑖≥𝑘−1 𝑖−𝑘 𝑖−𝑘+1 𝑛,𝑖, where defining 𝑐−1 =0. Conversely, starting from the sequences given. Hence another determinant expression for the Bessel ∞ 𝑛 ∞ 𝑛 𝑐(𝑡) =∑ 𝑐𝑛𝑡 𝑟(𝑡) =∑ 𝑟𝑛𝑡 polynomials is given by 𝑐𝑛𝑦𝑛(2𝑧) =𝑛 𝐻 (𝑧) with 𝐻𝑛(𝑧) = 𝑛=0 ,and 𝑛=0 ,theinfinitearray 𝑛(𝑛+1)/2 𝑛 𝑛 2 2 (𝑎𝑛,𝑘)𝑛,𝑘≥0 defined by above relations is an exponential Riordan det[𝑎𝑖+𝑗(𝑧)]0≤𝑖,𝑗≤𝑛,where𝑐𝑛 =(−1) 2 ∏𝑗=1(𝑗! /(2𝑗)! ) 𝑘 𝑚 matrix. and 𝑎𝑘(𝑧) = ∑𝑚=0(𝑧 /(𝑘−𝑚)!). In this paper, using the production matrix of Bessel From this proposition, the production matrix of 𝐴= matrix which is represented as exponential Riordan array, we [𝑔(𝑡), ℎ(𝑡)] is 𝑃=(𝑝𝑖,𝑗)𝑖,𝑗≥0 with 𝑝𝑖,𝑗 = (𝑖!/𝑗!)(𝑐𝑖−𝑗 +𝑗𝑟𝑖−𝑗+1) show that a variation of the generalized Bessel polynomial (𝑐−1 =0), where the numbers 𝑐𝑛 and 𝑟𝑛 are determined by ∞ 𝑛 󸀠 ∞ 𝑛 sequence is of Sheffer type, and we give a determinant 𝑐(𝑡) =∑𝑛=0 𝑐𝑛𝑡 =𝑔(𝑓(𝑡))/𝑔(𝑓(𝑡)), 𝑟(𝑡)𝑛=0 =∑ 𝑟𝑛𝑡 = formula for the generalized Bessel polynomials. As a result, 󸀠 𝑓 (𝑓(𝑡)).Thesequence(𝑐𝑛)𝑛≥0 and (𝑟𝑛)𝑛≥0 are called, respec- the Bessel polynomial is represented as determinant the tively, the 𝑐-sequence and the 𝑟-sequence of exponential entries of which involve Catalan numbers. Riordan array 𝐴 = [𝑔(𝑡), ℎ(𝑡)]. 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): ∞ 𝑛 ∞ 𝑛 𝑔(𝑡) = 1+ ∑𝑛=1 𝑔𝑛(𝑡 /𝑛!), ℎ(𝑡) = ∑𝑛=1 ℎ𝑛(𝑡 /𝑛!) with ℎ =0̸ ∞ 𝑡𝑛 1 1 be two formal power series. An exponential Riordan ∑𝑠 (𝑥) = 𝑒𝑥𝑓(𝑡), 𝐴=(𝑎 ) 𝑛 (5) array is an infinite lower triangular array 𝑛,𝑘 𝑛,𝑘≥0 𝑛=0 𝑛! 𝑔(𝑓 (𝑡)) whose 𝑘th column has exponential generating function 𝑘 (1/𝑘!)𝑔(𝑡)ℎ(𝑡) . The array corresponding to the pair 𝑔(𝑡), where 𝑓(𝑡) is a delta series and 𝑔(𝑡) an invertible series, ℎ(𝑡) 𝐴 = [𝑔(𝑡), ℎ(𝑡)] is denoted by .Thegrouplawis and 𝑓(𝑡) is the compositional inverse of 𝑓(𝑡).Thesequence [𝑔(𝑡), ℎ(𝑡)][𝑑(𝑡), 𝑙(𝑡)] = [𝑔(𝑡)𝑑(ℎ(𝑡)), 𝑙(ℎ(𝑡))] then given by . 𝑠𝑛(𝑥) in (5) is called the Sheffer sequence for the pair From the definition it follows at once that the bivariate (𝑔(𝑡), 𝑓(𝑡)).TheSheffersequence,𝑝𝑛(𝑥) for (1, 𝑓(𝑡)) is called 𝐴 = [𝑔(𝑡), ℎ(𝑡)] 𝐺 (𝑥, 𝑡) = generating function of ,namely, 𝐴 theassociatedsequenceanditsatisfiesbinomialformula ∑ 𝑎 𝑥𝑘(𝑡𝑛/𝑛!) 𝑛 𝑛,𝑘 𝑛,𝑘 ,isgivenby 𝑝𝑛(𝑥+𝑦)=𝑘=0 ∑ 𝑝𝑘(𝑥)𝑝𝑛−𝑘(𝑦),sowealsosaythatitis of binomial type. In [13],theauthorprovedthefollowing theorem ([13,Theorem2.4]). 𝐺𝐴 (𝑥,) 𝑡 =𝑔(𝑡) exp (𝑥𝑓 (𝑡)) . (4) Lemma 2. Let [𝑔(𝑡), 𝑓(𝑡)] be an exponential Riordan array 𝐴 = [𝑔(𝑡), ℎ(𝑡)] Each exponential Riordan array is asso- with the 𝑐-sequences (𝑐𝑖)𝑖≥0 and 𝑟-sequences (𝑟𝑖)𝑖≥0.Let 𝑃 ciated a matrix called its production matrix [12], which is (𝑎𝑛(𝑥))𝑛≥0 be the Sheffer polynomial sequence for (𝑔(𝑡), 𝑓(𝑡)). −1 determined by 𝑃=𝐴 𝐴,where𝐴 is the matrix 𝐴 with its Then 𝑎0(𝑥) =, 1 𝑎1(𝑥)=𝑥−𝑐0,andforall𝑛≥0, 𝑎𝑛+1(𝑥) = top row removed. The production matrix of an exponential det(𝑥𝐼𝑛+1−𝑃𝑛+1),where𝑃𝑛+1 is the (𝑛+1)th principal submatrix Riordan array is characterized by the following lemma [10]. of the production matrix 𝑃. Journal of Applied Mathematics 3

3. Bessel Polynomials and Bessel Matrices Proof. From Theorem 3,weobtain

𝑛≥𝑘≥0 ∞ 𝑛 ∞ 𝑛 For , the signless Bessel number of the 𝑡 𝑛 𝑥 𝑡 𝑛−𝑘 ∑ 𝑝 (𝑧, 𝑥) = ∑ (2𝑧) 𝑝 ( ) first kind 𝑎(𝑛, 𝑘) is defined to be the coefficient of 𝑧 in 𝑛 𝑛 𝑛=0 𝑛! 𝑛=0 2𝑧 𝑛! Bessel polynomial 𝑦𝑛−1(𝑧) with 𝑎(0, 0). =1 The infinite lower triangular matrix 𝐴 with generic entry 𝑎(𝑛, 𝑘) is called ∞ 𝑥 (2𝑧𝑡)𝑛 = ∑ 𝑝 ( ) =𝑒𝑥((1−√1−4𝑧𝑡)/2𝑧). signless Bessel matrix of the first kind. From [4], the signless 𝑛 2𝑧 𝑛! Bessel matrix of the first kind can be represented as the 𝑛=0 exponential Riordan array 𝐴=[1,1−√1−2𝑡]. (10) 𝑛 Let 𝑝𝑛(𝑥) = 𝑥 𝑦𝑛−1(1/𝑥) for 𝑛≥1and 𝑝0(𝑥) =.Then 1 Using (5), we obtain the desired result. the coefficients matrix of polynomial sequence {𝑝𝑛(𝑥)}𝑛≥0 is the signless Bessel matrix of the first kind 𝐴=[1,1−√1−2𝑡]. 𝑛 From the previous proof, the polynomial sequence Let 𝜃𝑛(𝑥) = 𝑥 𝑦𝑛(1/𝑥) for 𝑛≥0.Then𝜃𝑛(𝑥) = (1/𝑥)𝑝𝑛+1(𝑥), 𝑝 (𝑧, 𝑥) 𝑛 𝑛 𝑘 𝑛 has exponential generating function and 𝜃𝑛(𝑥) = 𝑥 𝑦𝑛(1/𝑥) = ∑𝑘=0 𝑎(𝑛+1,𝑘+1)𝑥.By[10], the infinite lower triangular matrix 𝐵 with generic entry ∞ 𝑡𝑛 ∑𝑝 (𝑧, 𝑥) =𝑒𝑥((1−√1−4𝑧𝑡)/2𝑧), 𝐵𝑛,𝑘 = 𝑎(𝑛+1,𝑘+1)is the exponential Riordan array 𝐵= 𝑛 (11) 𝑛=0 𝑛! [1/√1−2𝑡,1−√1−2𝑡].WecallthismatrixtheBesselmatrix since it is the coefficients matrix of the Bessel polynomials. and 𝑝𝑛(𝑥) =𝑛 𝑝 (1/2, 𝑥).So{𝑝𝑛(𝑧, 𝑥)}𝑛≥0 is a generalization of From (4), one has {𝑝𝑛(𝑥)}𝑛≥0. Instead of working with 𝑦𝑛(𝑧, 𝑎,,itissomewhateas- 𝑏) ∞ 𝑛 𝑡 𝑥(1−√1−2𝑡) ier to work with the polynomials 𝑠𝑛(𝑧,𝑎,𝑥) defined as ∑𝑝𝑛 (𝑥) =𝑒 , (6) 𝑛 𝑛 𝑛+𝑘+𝑎−2 𝑘 𝑛−𝑘 𝑛! 𝑠𝑛(𝑧, 𝑎, 𝑥) = ∑𝑘=0 ( 𝑘 )( 𝑘 )𝑘!𝑧 𝑥 ,whicharerelated 𝑛=0 𝑛 to the 𝑦𝑛(𝑧, 𝑎, 𝑏) by 𝑠𝑛(𝑧, 𝑎, 𝑥) =𝑥 𝑦𝑛(𝑧, 𝑎,, 𝑥) 𝑛≥0.Itis ∞ 𝑛 𝑡 1 𝑥(1−√1−2𝑡) easy to check that 𝑠𝑛(1/2, 2, 𝑥) =𝑛 𝜃 (𝑥).So{𝑠𝑛(𝑧, 𝑎, 𝑛≥0𝑥)} is a ∑𝜃𝑛 (𝑥) = 𝑒 . (7) {𝜃𝑛(𝑥)}𝑛≥0 𝑛=0 𝑛! √1−2𝑡 generalization of .

Theorem 5. {𝑠𝑛(𝑧, 𝑎, 𝑛≥0𝑥)} is a Sheffer polynomial sequence Henceweobtainthefollowingresults(seealso[14, 16]). in the variable 𝑥. Theorem 3. {𝑝 (𝑥)} 𝑎−2 The polynomial sequence 𝑛 𝑛≥0 is of bino- Proof. Let 𝐵(𝑧) = [(1/√1 − 4𝑧𝑡)((1 − √1−4𝑧𝑡)/2𝑧𝑡) ,(1− {𝜃 (𝑥)} mial type, and the polynomial sequence 𝑛 𝑛≥0 is a Sheffer √1−4𝑧𝑡)/2𝑧].By(5.72) of [17,page203],one sequence. has the general term of 𝐵(𝑧) which is 𝐵(𝑧)𝑛,𝑘 = [𝑡𝑛/𝑛!](1/√1−4𝑧𝑡)𝑡𝑘((1 − √1−4𝑧𝑡)/2𝑧𝑡)𝑘+𝑎−2 = The generalized Bessel polynomial 𝑦𝑛(𝑧, 𝑎, 𝑏) was defined [𝑡𝑛/𝑛!]𝑡𝑘 ∑∞ ( 2𝑛+𝑘+𝑎−2 )(𝑧𝑡)𝑛 =(𝑛 )(2𝑛−𝑘+𝑎−2 )(𝑛−𝑘)!𝑧𝑛−𝑘 by Krall and Frink [2] as the polynomial solution of the 𝑛=0 𝑛 𝑘 𝑛−𝑘 . 𝑛 𝑘 𝑛 𝑛 2𝑛−𝑘+𝑎−2 differential equation Hence ∑𝑘=0 𝐵(𝑧)𝑛,𝑘𝑥 = ∑𝑘=0 ( 𝑘 )( 𝑛−𝑘 )(𝑛 − 𝑛−𝑘 𝑘 𝑛 𝑛 2𝑛−𝑘+𝑎−2 𝑛−𝑘 𝑘 𝑘)!𝑧 𝑥 = ∑𝑘=0 ( 𝑛−𝑘 )( 𝑛−𝑘 )(𝑛 − 𝑘)!𝑧 𝑥 = 2 󸀠󸀠 󸀠 𝑛 𝑛 𝑛+𝑘+𝑎−2 𝑘 𝑛−𝑘 𝑛 𝑛 𝑛 𝑛+𝑘+𝑎−2 𝑘 𝑧 𝑦𝑛 (𝑧) + (𝑎𝑧) +𝑏 𝑦𝑛 (𝑧) =𝑛(𝑛+𝑎−1) 𝑦𝑛 (𝑧) , ∑𝑘=0( 𝑘 )( 𝑘 )𝑘!𝑧 𝑥 =𝑥 ∑𝑘=0( 𝑘 )( 𝑘 )𝑘!(𝑧/𝑥) = 𝑛 𝑥 𝑦𝑛(𝑧, 𝑎, 𝑥)𝑛 =𝑠 (𝑧, 𝑎,. 𝑥) Therefore, the exponential gener- 𝑏 =0,̸ 𝑎=0,−1,−2,....̸ ating function of the sequence {𝑠𝑛(𝑧, 𝑎, 𝑛≥0𝑥)} is (8) ∞ 𝑡𝑛 1 1−√1−4𝑧𝑡 𝑎−2 ∑𝑠 (𝑧, 𝑎, 𝑥) = ( ) The explicit expression of the generalized Bessel polynomials 𝑛 𝑛! √1−4𝑧𝑡 2𝑧𝑡 is 𝑛=0 (12) ×𝑒𝑥((1−√1−4𝑧𝑡)/2𝑧). 𝑛 𝑛 𝑛+𝑘+𝑎−2 𝑧 𝑘 𝑦 (𝑧, 𝑎, 𝑏) = ∑ ( )( )𝑘!( ) . 𝑛 𝑘 𝑘 𝑏 (9) 𝑘=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 (𝑛 + 𝑎 − 1)(2𝑛 +𝑛+1 𝑎−2)𝑦 (𝑧) = [(2𝑛 + 𝑎)(2𝑛 + Theorem 6. The sequences {𝑝𝑛(𝑧, 𝑥)}𝑛≥0 and {𝑠𝑛(𝑧, 𝑎, 𝑛≥0𝑥)} 𝑎−2)(𝑧/𝑏)+𝑎−2](2𝑛+𝑎−1)𝑦𝑛(𝑧) + 𝑛(2𝑛𝑛−1 +𝑎)𝑦 (𝑧), 𝑛≥2, satisfy the following convolution formulas: with initial conditions 𝑦0(𝑧) = 1 and 𝑦1(𝑧) = 1 + 𝑎(𝑧/𝑏). We introduce a two-variable polynomial sequence 𝑛 𝑝 (𝑧, 𝑥) 𝑝 (𝑧, 𝑥) = (2𝑧)𝑛𝑝 (𝑥/2𝑧) 𝑝 (𝑧, 𝑥) = 𝑛 by 𝑛 𝑛 .Then 𝑛 𝑝𝑛 (𝑧, 𝑥 + 𝑦) = ∑𝑝𝑘 (𝑧, 𝑥) 𝑝𝑛−𝑘 (𝑧, 𝑦) , 𝑛 𝑛 𝑛−𝑘 𝑛−𝑘 𝑘 𝑛−1 2𝑛−𝑘−1 𝑛−𝑘 𝑘 𝑘=0 ∑𝑘=0 𝑎(𝑛, 𝑘)2 𝑧 𝑥 = ∑𝑘=0 ( 𝑘−1 )( 𝑛−1 )(𝑛−𝑘)!𝑧 𝑥 . (13) 𝑛 Theorem 4. {𝑝 (𝑧, 𝑥)} 𝑛 𝑛≥0 is a sequence of polynomials of bino- 𝑠𝑛 (𝑧, 𝑎, 𝑥 +𝑦)= ∑𝑠𝑘 (𝑧, 𝑎, 𝑥) 𝑝𝑛−𝑘 (𝑧, 𝑦) . mial type in the variable 𝑥. 𝑘=0 4 Journal of Applied Mathematics

Note that the coefficients matrix of the polynomial √1 − 4𝑡)/2𝑡.Theformulae(11)and(12) imply the connection √ sequence {𝑝𝑛(𝑧, 𝑥)}𝑛≥0 is 𝐴(𝑧) = [1, (1− 1−4𝑧𝑡)/2𝑧], betweenthegeneralizedBesselpolynomialsandCatalan which is a generalization of the signless Bessel matrix of numbers. In the following we will give some determinant the first kind 𝐴=[1,1−√1−2𝑡] while the coeffi- expressions for Bessel polynomials, in which the entries cients matrix of the polynomial sequence {𝑠𝑛(𝑧, 𝑎, 𝑛≥0𝑥)} involve Catalan numbers. 𝑎−2 Define 𝑃(𝑧, 𝑎) to be the infinite Hessenberg matrix with is 𝐵(𝑧, 𝑎) =√ [(1/ 1 − 4𝑧𝑡)((1 − √1−4𝑧𝑡)/2𝑧𝑡) ,(1 − general entry √1−4𝑧𝑡)/2𝑧], which is a generalization of the Bessel matrix 𝐵=[1/√1−2𝑡,1−√1−2𝑡]. {1, if 𝑖=𝑗−1, { 𝑖! 𝑝 (𝑧,) 𝑎 = − (2𝑖 + 𝑎) 𝐶 𝑧𝑖−𝑗+1, 𝑖≥𝑗≥0, 4. Determinant Formulae for 𝑖,𝑗 { 𝑗! 𝑖−𝑗 if (14) Bessel Polynomials { {0, otherwise. 2𝑛 The Catalan numbers, 𝐶𝑛 = (1/(𝑛 + 1)) ( 𝑛 ), may be defined ∞ 𝑛 by the ordinary generating function ∑𝑛=0 𝐶𝑛𝑡 =(1− The first rows of 𝑃 are

−𝑎𝐶0𝑧1 0 0⋅⋅⋅ 2 − (2+𝑎) 𝐶1𝑧 − (2+𝑎) 𝐶0𝑧1 0⋅⋅⋅ 3 2 ( −2 (4+𝑎) 𝐶2𝑧 −2 (4+𝑎) 𝐶1𝑧 − (4+𝑎) 𝐶0𝑧1⋅⋅⋅) ( 4 3 2 ) 𝑃 (𝑧,) 𝑎 = ( −6 (6+𝑎) 𝐶3𝑧 −6 (6+𝑎) 𝐶2𝑧 −3 (6+𝑎) 𝐶1𝑧 − (6+𝑎) 𝐶0𝑧⋅⋅⋅) . ( 5 4 3 2 ) (15) −24 (8+𝑎) 𝐶4𝑧 −24 (8+𝑎) 𝐶3𝑧 −12 (8+𝑎) 𝐶2𝑧 −4 (8+𝑎) 𝐶1𝑧 ⋅⋅⋅ ...... d ( )

Theorem 7. For 𝑛 = 0, 1, 2, . ., the polynomial 𝑠𝑛+1(𝑧, tion 𝑠𝑛+1(𝑧,𝑎,𝑥) = det (𝑥𝐼𝑛+1 −𝑃𝑛+1(𝑧,𝑎)),that 𝑎, 𝑥) admits the following determinant representa- is,

󵄨 𝑥+𝑎𝐶 𝑧 −1 0 ⋅⋅⋅ 0 󵄨 󵄨 0 󵄨 󵄨 2 󵄨 󵄨 (2+𝑎) 𝐶1𝑧 𝑥+(2+𝑎) 𝐶0𝑧 −1 ⋅⋅⋅ 0 󵄨 󵄨 3 2 󵄨 󵄨 2 (4+𝑎) 𝐶2𝑧 2 (4+𝑎) 𝐶1𝑧 𝑥+(4+𝑎) 𝐶0𝑧 ⋅⋅⋅ 0 󵄨 𝑠 (𝑧, 𝑎, 𝑥) = 󵄨 󵄨 , 𝑛+1 󵄨 . . . . 󵄨 (16) 󵄨 . . . d . 󵄨 󵄨 . . . . 󵄨 󵄨𝑛! (2𝑛 + 𝑎) 𝑛! (2𝑛 + 𝑎) 𝑛! (2𝑛 + 𝑎) 󵄨 󵄨 𝐶 𝑧𝑛+1 𝐶 𝑧𝑛 𝐶 𝑧𝑛−1 ⋅⋅⋅ 𝑥+(2𝑛 + 𝑎) 𝐶 𝑧󵄨 󵄨 0! 𝑛 1! 𝑛−1 2! 𝑛−2 0 󵄨

where 𝑃𝑛+1(𝑧, 𝑎) is the (𝑛 + 1)th principal submatrix of The generating functions for the 𝑟-sequence and 𝑐- 󸀠 Hessenberg matrix 𝑃(𝑧, 𝑎) defined by (14). sequence of [𝑔(𝑡), 𝑓(𝑡)] are 𝑟(𝑡) = 𝑓 (𝑓(𝑡)) = √1−4𝑧𝑡= 1− ∞ 𝑛 𝑛 󸀠 2∑𝑛=1 𝐶𝑛−1𝑧 𝑡 , 𝑐(𝑡) = 𝑔 (𝑓(𝑡))/𝑔(𝑓(𝑡)) = (2 − 𝑎)𝑧𝐶(𝑧𝑡) − 𝑔(𝑡) = (1 − 2𝑧𝑡)(1 −𝑧𝑡)𝑎−2 𝑓(𝑡) = 𝑡(1 −𝑧𝑡) √ ∞ 𝑛+1 𝑛 Proof. Let and ; 2𝑧/( 1−4𝑧𝑡)= ∑𝑛=0 −(2𝑛 + 𝑎)𝐶𝑛𝑧 𝑡 .ByLemma 1,the then 𝑓(𝑡) = (1−√1−4𝑧𝑡)/2𝑧, 1/𝑔(𝑓(𝑡)) = √(1/ 1 − 4𝑧𝑡)((1− production matrix of [𝑔(𝑡), 𝑓(𝑡)] is exactly the Hessenberg √ 𝑎−2 𝑃 1−4𝑧𝑡)/2𝑧𝑡) . Therefore, from (12), {𝑠𝑛(𝑧, 𝑎, 𝑛≥0𝑥)} is a matrix defined by (14). Hence, by Lemma 2,weobtainthe Sheffer polynomial sequence for the pair ((1 − 2𝑧𝑡)(1 − desired result. 𝑎−2 𝑧𝑡) , 𝑡(1 − 𝑧𝑡)). Corollary 8. For 𝑛 = 0, 1, 2, . .,onehas𝑦𝑛+1(𝑧, 𝑎, 𝑥) = det(𝐼𝑛+1 −(1/𝑥)𝑃𝑛+1(𝑧, 𝑎)).Particularly, Journal of Applied Mathematics 5

󵄨 󵄨 󵄨 2+2𝐶0𝑧 −1 0 ⋅⋅⋅ 0 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2! 󵄨 󵄨 2 󵄨 󵄨 ⋅2𝐶1𝑧 2+4𝐶0𝑧 −1 ⋅⋅⋅ 0 󵄨 󵄨 0! 󵄨 󵄨 󵄨 󵄨 3! 3! 󵄨 1 󵄨 ⋅2𝐶 𝑧3 ⋅2𝐶 𝑧2 2+6𝐶 𝑧 ⋅⋅⋅ 0 󵄨 𝑦 (𝑧) = 󵄨 2 1 0 󵄨 . 𝑛+1 2𝑛+1 󵄨 0! 1! 󵄨 (17) 󵄨 󵄨 󵄨 󵄨 󵄨 . . . . 󵄨 󵄨 . . . d . 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨(𝑛+1)! (𝑛+1)! (𝑛+1)! 󵄨 󵄨 ⋅2𝐶 𝑧𝑛+1 ⋅2𝐶 𝑧𝑛 ⋅2𝐶 𝑧𝑛−1 ⋅⋅⋅ 2+2(𝑛+1) 𝐶 𝑧󵄨 󵄨 0! 𝑛 1! 𝑛−1 2! 𝑛−2 0 󵄨

Corollary 9. For 𝑛 = 0, 1, 2, . .,wehave

󵄨 󵄨 󵄨 𝑥+𝐶0 −1 0 ⋅⋅⋅ 0 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2! 󵄨 󵄨 󵄨 󵄨 𝐶1 𝑥+2𝐶0 −1 ⋅⋅⋅ 0 󵄨 󵄨 0!2 󵄨 󵄨 󵄨 󵄨 3! 3! 󵄨 󵄨 𝐶 𝐶 𝑥+3𝐶 ⋅⋅⋅ 0 󵄨 𝜃 (𝑥) = 󵄨 2 2 1 0 󵄨 , 𝑛+1 󵄨 0!2 1!2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 . . . . 󵄨 󵄨 . . . d . 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨(𝑛+1)! (𝑛+1)! (𝑛+1)! 󵄨 󵄨 𝐶 𝐶 𝐶 ⋅⋅⋅ 𝑥+(𝑛+1) 𝐶 󵄨 󵄨 0!2𝑛 𝑛 1!2𝑛−1 𝑛−1 2!2𝑛−2 𝑛−2 0󵄨 (18) 󵄨 󵄨 󵄨 1+𝐶0𝑥 −𝑥 0 ⋅⋅⋅ 0 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2! 󵄨 󵄨 󵄨 󵄨 𝐶1𝑥1+2𝐶0𝑥 −𝑥 ⋅⋅⋅ 0 󵄨 󵄨 0!2 󵄨 󵄨 󵄨 󵄨 3! 3! 󵄨 󵄨 𝐶 𝑥 𝐶 𝑥1+3𝐶𝑥 ⋅⋅⋅ 0 󵄨 𝑦 (𝑥) = 󵄨 2 2 1 0 󵄨 . 𝑛+1 󵄨 0!2 1!2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 . . . . 󵄨 󵄨 . . . d . 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨(𝑛+1)! (𝑛+1)! (𝑛+1)! 󵄨 󵄨 𝐶 𝑥 𝐶 𝑥 𝐶 𝑥 ⋅⋅⋅ 1+(𝑛+1) 𝐶 𝑥󵄨 󵄨 0!2𝑛 𝑛 1!2𝑛−1 𝑛−1 2!2𝑛−2 𝑛−2 0 󵄨

Examples. (1) One has (2) One has

𝑦3 (𝑧, 𝑎, 𝑥) 󵄨 󵄨 󵄨 𝑥+𝑎𝐶0𝑧−10󵄨 1 󵄨 2 󵄨 𝑠3 (𝑧, 𝑎, 𝑥) =( ) 󵄨 (2+𝑎) 𝐶 𝑧 𝑥+(2+𝑎) 𝐶 𝑧−1󵄨 𝑥3 󵄨 1 0 󵄨 󵄨2 (4+𝑎) 𝐶 𝑧3 2 (4+𝑎) 𝐶 𝑧2 𝑥+(4+𝑎) 𝐶 𝑧󵄨 󵄨 𝑥+𝑎𝐶 𝑧−10󵄨 󵄨 2 1 0 󵄨 󵄨 0 󵄨 󵄨 2 󵄨 2 = 󵄨 (2+𝑎) 𝐶1𝑧 𝑥+(2+𝑎) 𝐶0𝑧−1󵄨 𝑧 𝑧 󵄨 3 2 󵄨 =1+3(𝑎+2) ( )+3(𝑎+2)(𝑎+3) ( ) 󵄨2 (4+𝑎) 𝐶2𝑧 2 (4+𝑎) 𝐶1𝑧 𝑥+(4+𝑎) 𝐶0𝑧󵄨 (19) 𝑥 𝑥 =𝑥3 +3(𝑎+2) 𝑧𝑥2 +3(𝑎+2)(𝑎+3) 𝑧2𝑥 𝑧 3 + (𝑎+2)(𝑎+3)(𝑎+4) ( ) . 𝑥 3 + (𝑎+2)(𝑎+3)(𝑎+4) 𝑧 . (20) 6 Journal of Applied Mathematics

(3) Consider the following: [11] T.-X. He and R. Sprugnoli, “Sequence characterization of 󵄨 󵄨 Riordan arrays,” Discrete Mathematics,vol.309,no.12,pp.3962– 󵄨2+2𝑧 −1 0 󵄨 1 󵄨 󵄨 3974, 2009. 𝑦 (𝑧) =( ) 󵄨 4𝑧2 2+4𝑧 −1 󵄨 3 3 󵄨 󵄨 [12] P. Peart and W.-J. Woan, “Generating functions via Hankel and 2 󵄨 3 2 󵄨 󵄨 24𝑧 12𝑧 2+6𝑧󵄨 (21) Stieltjes matrices,” Journal of Integer Sequences,vol.3,no.2, article 00.2.1, 2000. 2 3 = 1 + 6𝑧 + 15𝑧 + 15𝑧 . [13] S.-l. Yang, “Recurrence relations for the Sheffer sequences,” Linear Algebra and Its Applications,vol.437,no.12,pp.2986– (4) Consider the following: 2996, 2012. 󵄨 󵄨 󵄨𝑥+1 −1 0 󵄨 [14] S. Khan, M. W. Al-Saad, and G. Yasmin, “Some properties of 󵄨 󵄨 Hermite-based Sheffer polynomials,” Applied Mathematics and 𝜃 (𝑥) = 󵄨 1 𝑥+2 −1 󵄨 3 󵄨 󵄨 Computation,vol.217,no.5,pp.2169–2183,2010. 󵄨 3 3 𝑥+3󵄨 󵄨 󵄨 (22) [15] S. Roman, The Umbral Calculus, vol. 111, Academic Press, New =𝑥3 +6𝑥2 + 15𝑥 + 15. York, NY, USA, 1984. [16] L. Carlitz, “Anote on the Bessel polynomials,” Duke Mathemat- (5) One has ical Journal,vol.24,pp.151–162,1957. [17]R.L.Graham,D.E.Knuth,andO.Patashnik,Concrete Mathe- 󵄨 󵄨 󵄨1+𝑥 −𝑥 0 󵄨 matics, Addison-Wesley, New York, NY,USA, 2nd edition, 1994. 󵄨 󵄨 𝑦3 (𝑥) = 󵄨 𝑥1+2𝑥−𝑥󵄨 󵄨 󵄨 󵄨 3𝑥 3𝑥 1 +3𝑥󵄨 (23) = 15𝑥3 + 15𝑥2 +6𝑥+1.

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

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