Hindawi Publishing Corporation Journal of Discrete Mathematics Volume 2013, Article ID 450481, 10 pages http://dx.doi.org/10.1155/2013/450481

Research Article 𝑞-Pascal and 𝑞-Wronskian Matrices with Implications to 𝑞-Appell Polynomials

Thomas Ernst

Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden

Correspondence should be addressed to Thomas Ernst; [email protected]

Received 19 June 2012; Accepted 2 December 2012

Academic Editor: Franck Petit

Copyright © 2013 Thomas Ernst. 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.

Weintroduce a 𝑞-deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for 𝑞-Appellpolynomials,andinparticularfor𝑞-Bernoulli and 𝑞-Euler polynomials. Furthermore, the 𝑞-𝐿-polynomial, anticipated by Ward, can be expressed as a sum of products of 𝑞-Bernoulli and 𝑞-Euler polynomials. The pseudo 𝑞-Appell polynomials, which are first presented in this paper, enable multiple 𝑞-analogues of the Yang and Youn formulas. The generalized 𝑞-Pascal functional matrix, the 𝑞-Wronskian vector of a function, and the vector of 𝑞-Appell polynomials together with the 𝑞-deformed 𝜖 matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of 𝑞-Appell numbers, improving on Al-Salam 1967. Finally, we find a 𝑞-difference equation for the 𝑞-Appell polynomial of degree 𝑛.

1. Introduction we will need pseudo 𝑞-Appell polynomials, which correspond to the other basic 𝑞-addition, JHC; these polynomials were In this paper we will introduce the 𝑞-Pascal and 𝑞-Wronskian first seen in5 [ ]. The 𝑞-Appell numbers form a certain matrices in a general setting, with the aim of fruitful applica- algebraicstructure,withtwooperations,whichwasfirst tions for 𝑞-Appell polynomials. These two matrices contain described by Al-Salam [6]. This structure will in a certain acertain𝑞-difference operator, just like the 𝑞-deformed way be generalized to 𝑞-Appell polynomials. The 𝑞-Bernoulli Leibniz functional matrix from [1]. By the 𝑞-Leibniz formula, and 𝑞-Euler polynomials are presented along with the 𝑞- products of such matrices contain a certain operator ⋅𝜖,justas Lucas polynomials; these definitions coincide with those in the previous article, but with a slightly different definition. given in [7]. In Section 3 we come to the core of the paper, However,sinceinalmostallequationswecomputethe which starts with the 𝑞-Pascal matrix and the 𝑞-Wronskian function value at 𝑡=0, this operator will convert to ordinary vector. In the beginning, we briefly pass to the shorter paper matrix multiplication by formula (6). There is a connection to [8], to find a few 𝑞-analogues of truncated 𝑞-exponential the 𝑞-Pascal matrix [2]forthespecialcase𝑓=𝑞-exponential generating function formulas. We mostly follow the ordering function. The Appell polynomials are seldom encountered in in [9]. Sometimes multiple 𝑞-analogues follow, and these the literature; one exception was Carlitz’ article [3], where will usually come with the pseudo 𝑞-Appell polynomials. aformulafortheproductoftwogeneratingfunctionswas A couple of general formulas for 𝑞-Appell polynomials are given. Carlitz cleverly observed that the generating function given; sometimes these are specialized to 𝑞-Bernoulli and 𝑞- forAppellpolynomialscanbeinvertedwhen𝑓(0) = 1.The Eulerpolynomials.Inthelastpart,wefinda𝑞-difference NWA 𝑞-addition seldom occurs in the literature, except for equation for general 𝑞-Appell polynomials. In an appendix we the author’s papers; it was first seen in an Italian paper by Nalli compare our formulas for Bernoulli and Euler polynomials [4]. Corresponding to the two dual 𝑞-additions, NWA and with Hansen’s table [10], where another notation is used. JHC, there are two dual 𝑞-Bernoulli polynomials, with the We now start with the definition of the umbral method same names; these are special cases of 𝑞-Appell polynomials, from the recent book [11]. Some of the notations are well the content of Section 2. It turns out that for certain purposes known and will be skipped. 2 Journal of Discrete Mathematics

Definition 1. Let the Gauss 𝑞-binomial coefficient be defined We make the convention that all 𝑛×𝑛matrices are denoted by by the first index 𝑛. The rows and columns are denoted by 𝑖 and 𝑗.The𝑞-Pascal matrix [2] is defined by 𝑛 {𝑛}𝑞! ( ) ≡ , 𝑘=0,1,...,𝑛. (1) 𝑖 𝑘 {𝑘} !{𝑛−𝑘} ! 𝑃 (𝑥) (𝑖, 𝑗) ≡( ) 𝑥𝑖−𝑗. 𝑞 𝑞 𝑞 𝑛,𝑞 𝑗 (8) 𝑞 𝑎 𝑏 Let and be any elements with commutative multiplication. 𝐻 Then the NWA 𝑞-addition is given by The Polya-Vein matrix 𝑛,𝑞 [2]isgivenby 𝑛 𝐻 (𝑖, 𝑖) −1 ≡ {𝑖} , 𝑖=1,...,𝑛−1, 𝑛 𝑛 𝑛,𝑞 𝑞 (𝑎⊕ 𝑏) ≡ ∑( ) 𝑎𝑘𝑏𝑛−𝑘, 𝑛=0,1,2,.... 𝑞 𝑘 (2) (9) 𝑘=0 𝑞 𝐻𝑛,𝑞 (𝑖, 𝑗) ≡ 0,𝑗 =𝑖−1.̸ The JHC 𝑞-addition is the function: 2. 𝑞-Appell Polynomials 𝑛 𝑛 𝑛 ( 𝑘 ) (𝑎 ⊞ 𝑏) ≡ ∑( ) 𝑞 2 𝑏𝑘𝑎𝑛−𝑘 𝑞 𝑘 The Appell polynomials have been around for more than a 𝑘=0 𝑞 century, but this general notion is still not so well known. The (3) 𝑏 usual definition is =𝑎𝑛(− ;𝑞) , 𝑛=0,1,2,.... ∞ 𝑡𝜈 𝑎 𝑛 𝑓 (𝑡) 𝑒𝑥𝑡 = ∑ Φ (𝑥) ,𝑓(𝑡) ∈ C [[𝑥]] . 𝜈! 𝜈 (10) The 𝑞-exponential function is defined by 𝜈=0 ∞ The original article by Appell [15] started with the definition 1 𝑘 (𝑧) ≡ ∑ 𝑧 . 𝑑𝐴 𝑛/𝑑𝑥=𝑛𝐴𝑛−1. Then Appell showed that multiplication of E𝑞 {𝑘} ! (4) 𝑘=0 𝑞 two Appell polynomials is commutative. He then considered the inverse of 𝐴𝑛 as a natural generalization of ordinary divi- The 𝑞-derivative is defined by sion. The 𝑞-Appell polynomials form a natural generalization of the Appell polynomials. {𝑓 (𝑥) −𝑓(𝑞𝑥) { , if 𝑞∈C \ {1} ,𝑥=0,̸ { (1 − 𝑞) 𝑥 { 2.1. Definitions. We consider a certain value of the order (=1) { { for the 𝑞-Appell polynomials in [11]. The degree is only a { 𝑑𝑓 ( 𝑓) (𝑥) ≡ name;itisnotthedegreeofpolynomials. D𝑞 { (𝑥) if 𝑞=1, (5) {𝑑𝑥 { 𝑞 Φ (𝑥) 𝜈 { Definition 3. The -Appell polynomials 𝜈,𝑞 of degree { { 𝑑𝑓 are defined by (0) if 𝑥=0. {𝑑𝑥 ∞ 𝑡𝜈 𝑓 (𝑡) (𝑥𝑡) = ∑ Φ (𝑥) ,𝑓(𝑡) ∈ C [[𝑥]] . 𝑡 𝑦 𝑞 E𝑞 𝜈,𝑞 (11) Assume that , ,and are the variables. If we want to indicate 𝜈=0 {𝜈}𝑞! thevariableonwhichthe𝑞-difference operator is applied, we Φ (𝑥) 𝑞 denote the operator in (5)by(D𝑞,𝑡𝜑)(𝑡, 𝑦).Tosavespace,we We say that 𝜈,𝑞 is the -Appell polynomial sequence for 󸀠 𝑓(𝑡). sometimes write 𝑓 instead of D𝑞,𝑡𝑓. We put 𝑥=0and obtain Theorem 2. The following equation relates the 𝑛th 𝑞-difference ∞ 𝑡𝜈 𝑛 𝑓 (𝑡) = ∑ Φ ,Φ=1, operator at zero to the th derivative at zero for an analytic {𝜈} ! 𝜈,𝑞 0,𝑞 (12) function 𝑓(𝑧) [12], 𝜈=0 𝑞 where Φ𝜈,𝑞 is said to have degree 𝜈. {𝑛}𝑞! 𝑛𝑓 (0) = 𝑓(𝑛) (0) . (6) D𝑞 𝑛! 𝑞 Definition 4. The pseudo 𝑞-Appell polynomials Φ𝜈(𝑥) of 𝜈 For convenience, we will use the following two abbrevia- degree [5] are defined by (𝑘) tions for functions 𝑓(𝑡): 𝑓 represents the 𝑘th 𝑞-derivative ∞ 𝑡𝜈 𝑘 𝑓 (𝑡) (𝑥𝑡) = ∑ Φ𝑞 (𝑥) . of 𝑓 and 𝑓 represents the 𝑘th power of 𝑓.Wewillonly E1/𝑞 𝜈 (13) 𝜈=0 {𝜈}𝑞! consider functions in C[[𝑡]]. Sometimes functions can be 𝑞 given in other forms, we then mean the Taylor expansion of We say that Φ𝜈(𝑥) is the pseudo 𝑞-Appellpolynomialseq- the function around zero. In the sense of Milne-Thomson13 [ ] uence for 𝑓(𝑡). and Rainville [14]wewrite=̈ to indicate the symbolic nature. By putting 𝑥=0,wehave 𝑓(𝑥) ∈ C[𝑥] If , the ring of polynomials with complex ∞ 𝜈 coefficients, then the function 𝜖:C[𝑥] 󳨃→ C[𝑥] is defined 𝑡 𝑞 𝑓 (𝑡) = ∑ Φ𝜈, (14) by 𝜈=0 {𝜈}𝑞! 𝑞 𝑞 𝜖𝑓 (𝑥) ≡𝑓(𝑞𝑥). (7) where Φ𝜈 is called a Φ number of degree 𝜈. Journal of Discrete Mathematics 3

We have similarly Let us consider

𝑞 𝑞 𝑓𝑚 (𝑎𝑥) 𝑔𝑛 (𝑏𝑥) D𝑞Φ𝜈 (𝑥) = {𝜈}𝑞Φ𝜈−1,𝑞 (𝑞𝑥) , 𝑚 𝑛 𝜈 (15) 𝑚 𝑛 (20) Φ𝑞 (𝑥) =(Φ̈ ⊞ 𝑥) . = ∑ ∑( ) ( ) 𝑓 𝑔 𝑎𝑟𝑏𝑠𝑥𝑟+𝑠. 𝜈 𝑞 𝑞 𝑟 𝑠 𝑚−𝑟 𝑛−𝑠 𝑟=0 𝑠=0 𝑞 𝑞 2.2. 𝑞-Analogue of Carlitz’ Product Formula (1963). To moti- Now put 𝑞 vate the generalized -Pascal functional matrix, we show that ∞ 𝑡𝜈 the Carlitz [3,page247]formulafortheproductofgenerating ℎ(𝑡)−1 = ∑ Ψ󸀠 . {𝜈} ! 𝜈,𝑞 (21) functions for Appell polynomials has a 𝑞-analogue involving 𝜈=0 𝑞 󸀠 ∞ the NWA 𝑞-addition. Define the numbers {Φ } by the 𝜈,𝑞 𝜈=0 Because of (17)weget inverse of (12) (this is allowable, since 𝑓(0) = 1): 𝑓𝑚 (𝑎𝑥) 𝑔𝑛 (𝑏𝑥) ∞ 𝑡𝜈 𝑓(𝑡)−1 = ∑ Φ󸀠 ,Φ󸀠 =1. {𝜈} ! 𝜈,𝑞 0,𝑞 (16) 𝑚 𝑛 𝑚 𝑛 𝑟+𝑠 𝑟+𝑠 𝜈=0 𝑞 = ∑ ∑( ) ( ) 𝑓 𝑔 𝑎𝑟𝑏𝑠∑( ) Ψ󸀠 Ψ (𝑥) . 𝑟 𝑠 𝑚−𝑟 𝑛−𝑠 𝑗 𝑟+𝑠−𝑗,𝑞 𝑗,𝑞 𝑟=0 𝑠=0 𝑞 𝑞 𝑗=0 𝑞 Thenwehavethefollowing. (22) Theorem 5. 𝑞 An inverse relation formula for the -Appell Introduce the notation polynomials. 𝐴 (𝑗, 𝑞, 𝑚, 𝑛, 𝑎,𝑏) 𝑛 𝑛 𝑥𝑛 = ∑( ) Φ󸀠 Φ (𝑥) . 𝑚 𝑛 𝑘 𝑛−𝑘,𝑞 𝑘,𝑞 (17) 𝑚 𝑛 𝑟+𝑠 𝑟 𝑠 󸀠 𝑞 ≡ ∑ ∑( ) ( ) ( ) 𝑓 𝑔 𝑎 𝑏 Ψ , 𝑘=0 𝑟 𝑠 𝑗 𝑚−𝑟 𝑛−𝑠 𝑟+𝑠−𝑗,𝑞 𝑟=0 𝑠=0 𝑞 𝑞 𝑞 Proof. By the umbral definition of 𝑞-Appell polynomials we (23) have where 𝑛 𝑛 ∑( ) Φ󸀠 Φ (𝑥) 𝑚+𝑛 𝑘 𝑛−𝑘,𝑞 𝑘,𝑞 𝑘=0 𝑞 𝑓𝑚 (𝑎𝑥) 𝑔𝑛 (𝑏𝑥) = ∑ 𝐴 (𝑗, 𝑞, 𝑚,𝑗,𝑞 𝑛,𝑎,𝑏)Ψ (𝑥) . (24) 𝑗=0 𝑛 𝑛 𝑘 𝑘 = ∑( ) Φ󸀠 ∑( ) Φ 𝑥𝑙 Theorem 6. With the previously mentioned notation, one has 𝑘 𝑛−𝑘,𝑞 𝑙 𝑘−𝑙,𝑞 𝑘=0 𝑞 𝑙=0 𝑞 the following 𝑞-analogue of Carlitz [3,page247]: 𝑛 {𝑛} !Φ󸀠 𝑛 Φ ∞ ∞ 𝑢𝑚𝑣𝑛 𝑞 𝑛−𝑘,𝑞 𝑘−𝑙,𝑞 𝑙 ∑ ∑𝐴 (𝑗, 𝑞, 𝑚, 𝑛, 𝑎,𝑏) = ∑ ∑ 𝑥 (18) {𝑛−𝑘} ! {𝑙} !{𝑘−𝑙} ! 𝑚=0 𝑛=0 {𝑚}𝑞!{𝑛}𝑞! 𝑘=0 𝑞 𝑙=0 𝑞 𝑞 𝑘+𝑗 (25) 𝑛 󸀠 ∞ {𝑛} ! Φ Φ𝑡,𝑞 (𝑎𝑢𝑞 ⊕ 𝑏𝑣) = ∑ 𝑞 𝑥𝑙 ∑ 𝑠,𝑞 =𝑓(𝑢) 𝑔 (𝑣) ∑Ψ󸀠 . {𝑙} ! {𝑠} !{𝑡} ! 𝑘,𝑞 {𝑗} !{𝑘} ! 𝑙=0 𝑞 𝑠+𝑡=𝑛−𝑙 𝑞 𝑞 𝑘=0 𝑞 𝑞 𝑛 {𝑛} ! 𝑞 𝑙 𝑛 Proof. Thus we have = ∑ 𝑥 𝛿𝑛−𝑙 =𝑥 . {𝑙} ! ∞ ∞ 𝑢𝑚𝑣𝑛 𝑙=0 𝑞 ∑ ∑𝐴 (𝑗, 𝑞, 𝑚, 𝑛,𝑎,𝑏) {𝑚} !{𝑛} ! This completes the proof. 𝑚=0 𝑛=0 𝑞 𝑞 ∞ ∞ 𝑚 𝑛 𝑞 Ξ Ψ 𝑢𝑚𝑣𝑛 We introduce two other -Appell polynomials 𝑞 and 𝑞 = ∑ ∑ ∑ ∑ by 𝑚=0 𝑛=0 {𝑚}𝑞!{𝑛}𝑞! 𝑟=0 𝑠=0 ∞ 𝑡𝜈 𝑚 𝑛 𝑟+𝑠 𝑔 (𝑡) (𝑥𝑡) = ∑ Ξ (𝑥) ,𝑔(𝑡) ∈ C [[𝑥]] , =( ) ( ) ( ) 𝑓 𝑔 𝑎𝑟𝑏𝑠Ψ󸀠 E𝑞 𝜈,𝑞 𝑟 𝑠 𝑗 𝑚−𝑟 𝑛−𝑠 𝑟+𝑠−𝑗,𝑞 𝜈=0 {𝜈}𝑞! 𝑞 𝑞 𝑞 ∞ 𝑡𝜈 ∞ ∞ (𝑎𝑢)𝑟(𝑏𝑣)𝑠 𝑔 (𝑡) = ∑ Ξ𝜈,𝑞,Ξ0,𝑞 =1, = ∑ ∑ , 𝜈=0 {𝜈}𝑞! 𝑟=0 𝑠=0 {𝑟}𝑞!{𝑠}𝑞! (19) ∞ 𝑡𝜈 𝑟+𝑠 ∞ ∞ 𝑢𝑚𝑣𝑛 ℎ (𝑡) (𝑥𝑡) = ∑ Ψ (𝑥) ,ℎ(𝑡) ∈ C [[𝑥]] , ( ) Ψ󸀠 ∑ ∑𝑓 𝑔 E𝑞 𝜈,𝑞 𝑗 𝑟+𝑠−𝑗,𝑞 𝑚 𝑛 𝜈=0 {𝜈}𝑞! 𝑞 𝑚=0 𝑛=0 {𝑚}𝑞!{𝑛}𝑞! ∞ 𝑡𝜈 ∞ ∞ (𝑎𝑢)𝑟(𝑏𝑣)𝑠 ℎ (𝑡) = ∑ Ψ𝜈,𝑞,Ψ0,𝑞 =1. =𝑓(𝑢) 𝑔 (𝑣) ∑ ∑ , 𝜈=0 {𝜈}𝑞! 𝑟=0 𝑠=0 {𝑟}𝑞!{𝑠}𝑞! 4 Journal of Discrete Mathematics

𝑟+𝑠 ( ) Ψ󸀠 Proof. We prove (31)and(33) at the same time. We have in 𝑗 𝑟+𝑠−𝑗,𝑞 𝑞 the umbral sense

∞ 𝑘 (𝑎𝑢)𝑟(𝑏𝑣)𝑠 𝑓 (𝑡) ℎ (𝑡) = (𝑡 (Φ ⊕ Ψ )) . =𝑓(𝑢) 𝑔 (𝑣) ∑( ) Ψ󸀠 ∑ E𝑞 𝑞 𝑞 𝑞 (35) 𝑗 𝑘−𝑗,𝑞 {𝑟} !{𝑠} ! 𝑘=𝑗 𝑞 𝑟+𝑠=𝑘 𝑞 𝑞 Formulas (32)and(34)followbythecommutativityand 𝑘 ∞ the associativity of NWA. 𝑘 (𝑎𝑢𝑞 ⊕ 𝑏𝑣) =𝑓(𝑢) 𝑔 (𝑣) ∑( ) Ψ󸀠 . 𝑗 𝑘−𝑗,𝑞 {𝑘} ! 𝑘=𝑗 𝑞 𝑞 2.4. Specializations (26) Definition 11. The NWA 𝑞-Bernoulli polynomials (𝑥) 𝜈 BNWA,𝜈,𝑞 of degree are defined by 𝑞 2.3. Characterization of -Appell Numbers. We will now pre- 1 ∞ 𝜈 𝑞 𝑡 𝑡 B ,𝜈,𝑞 (𝑥) sent a characterization of -AppellnumberscomparewithAl- (𝑥𝑡) = ∑ NWA , |𝑡| <2𝜋. E𝑞 {𝜈} ! (36) Salam [6]. (E𝑞 (𝑡) −1) 𝜈=0 𝑞 A 𝑞 Definition 7. We denote by 𝑞 the set of all -Appell- The NWA 𝑞-Bernoulli polynomials are also given by numbers. Φ Ψ A 𝜈 Let 𝑛,𝑞 and 𝑛,𝑞 be two elements in 𝑞.Thenthe (𝑥) =(̈ ⊕ 𝑥) . operations + and ⋆ are defined as follows: BNWA,𝜈,𝑞 BNWA,𝑞 𝑞 (37)

(Φ +Ψ) ≡Φ +Ψ , The pseudo-𝑞-Bernoulli polynomials are defined by 𝑞 𝑞 𝑛 𝑛,𝑞 𝑛,𝑞 (27) 𝜈 𝑛 𝑞 𝑛 B (𝑥) =̈ (B ,𝑞 ⊞𝑞 𝑥) . (38) (Φ ⋆Ψ) ≡ ∑( ) Φ Ψ . NWA,𝜈 NWA 𝑞 𝑞 𝑛 𝑘 𝑛−𝑘,𝑞 𝑘,𝑞 (28) 𝑞 𝑘=0 𝑞 (𝑥) 𝜈 The JHC -Bernoulli polynomials BJHC,𝜈,𝑞 of degree Theorem 8. The identity element in A𝑞 with respect to ⋆ is are defined by 𝐼≡𝛿𝑛,0. This corresponds to 𝑓(𝑡) =1. 1 ∞ 𝑡𝜈 (𝑥) 𝑡 BJHC,𝜈,𝑞 E𝑞 (𝑥𝑡) = ∑ , |𝑡| <2𝜋. (39) Proof. Put 𝑘=𝑛and Φ𝑞 =1in (28). {𝜈} ! (E1/𝑞 (𝑡) −1) 𝜈=0 𝑞

Theorem 9 (Compare with [6,page34]). Let Φ𝑛,𝑞, Ξ𝑛,𝑞 and Ψ𝑛,𝑞 be three elements in A𝑞 corresponding to the generating 3. 𝑞-Pascal and 𝑞-Wronskian Matrices functions 𝑓(𝑡), 𝑔(𝑡),andℎ(𝑡),respectively.Furtherassumethat 𝑓(0) + ℎ(0) =0̸ .Then We now come to the most important part of the paper, which is of interest even for the case 𝑞=1,sincetheoriginalpaper Φ𝑛,𝑞 +Ψ𝑛,𝑞 ∈ A𝑞, by Yang and Youn is not very widespread. (29) 𝑞 Φ𝑛,𝑞 +Ψ𝑛,𝑞 has generating function 𝑓 (𝑡) +ℎ(𝑡) . Definition 12. The generalized -Pascal functional matrix is given by The associative law for addition reads: (P𝑛,𝑞) [𝑓 (𝑡, 𝑞)] (𝑖, 𝑗) Φ𝑛,𝑞 +(Ξ𝑛,𝑞 +Ψ𝑛,𝑞)=(Φ𝑛,𝑞 +Ξ𝑛,𝑞)+Ψ𝑛,𝑞. (30) { 𝑖 𝑖−𝑗 {( ) 𝑓(𝑡,𝑞) 𝑖≥𝑗, Proof. This follows from the definitions. { 𝑗 D𝑞,𝑡 if 𝑞 ≡{ { 𝑖,𝑗=0,1,2,...,𝑛−1. Theorem 10 (Compare with [6,page34]). Let Φ𝑛,𝑞, Ξ𝑛,𝑞,and { 0, , Ψ𝑛,𝑞 be three elements in A𝑞 corresponding to the generating { otherwise functions 𝑓(𝑡), 𝑔(𝑡),andℎ(𝑡),respectively.Then (40) Φ ⋆Ψ ∈ A , 𝑛,𝑞 𝑛,𝑞 𝑞 (31) In the special case 𝑓(𝑡, 𝑞) = constant,

Φ𝑛,𝑞 ⋆Ψ𝑛,𝑞 =Ψ𝑛,𝑞 ⋆Φ𝑛,𝑞, (32) (P𝑛,𝑞)[𝑓(𝑡,𝑞)]=𝑓(𝑡,𝑞)𝐼. (41)

Φ𝑛,𝑞 ⋆Ψ𝑛,𝑞 has generating function 𝑓 (𝑡) ℎ (𝑡) . (33) Definition 13. The 𝑞-deformed Wronskian vector is given by ⋆ The associative law for reads: 𝑖 (W𝑛,𝑞)[𝑓 (𝑡, 𝑞)] (𝑖,) 0 ≡ D𝑞,𝑡𝑓 (𝑡, 𝑞) , 𝑖=0,1,...,𝑛−1. Φ𝑛,𝑞 ⋆(Ξ𝑛,𝑞 ⋆Ψ𝑛,𝑞)=(Φ𝑛,𝑞 ⋆Ξ𝑛,𝑞)⋆Ψ𝑛,𝑞. (34) (42) Journal of Discrete Mathematics 5

Definition 14. The 𝑞-deformed Leibniz functional matrix is Example 17 (a 𝑞-analogue of [8, pages 231–234]). Let given by (G𝑛,𝑞) [𝑥] (𝑖, 𝑗) (L𝑛,𝑞) [𝑓 (𝑡, 𝑞)] (𝑖, 𝑗)

𝑖−𝑗𝑓(𝑡,𝑞) { 𝑖 {D𝑞,𝑡 {𝑔𝑖−𝑗 (𝑥) ( ) if 𝑖≥𝑗, { if 𝑖≥𝑗, { 𝑗 { {𝑖 − 𝑗} ! ≡ 𝑞 ≡ { 𝑞 { 𝑖,𝑗=0,1,2,...,𝑛−1. { 𝑖,𝑗=0,1,2,...,𝑛−1. { { { 0, otherwise, {0, otherwise, { (43) (53) In order to find a 𝑞-analogue of the formula [16] Put (L )[𝑓(𝑡) 𝑔 (𝑡)]=(L )[𝑓(𝑡)](L )[𝑔(𝑡)], 𝑛 𝑡𝑘 𝑛 𝑛 𝑛 (44) 𝑓 (𝑡, 𝑥, 𝑞) ≡ ∑𝑔 (𝑥) , 𝑘 {𝑘} ! (54) we infer that by the 𝑞-Leibniz formula [1] 𝑘=0 𝑞 (L𝑛,𝑞) [𝑓 (𝑡, 𝑞) 𝑔 (𝑡, 𝑞)] atruncated𝑞-exponential generating function of 𝑞-binomial (45) type. We find that =(L )[𝑓(𝑡,𝑞)]⋅ (L )[𝑔(𝑡,𝑞)], 𝑛,𝑞 𝜖 𝑛,𝑞 󵄨 P𝑛,𝑞 [𝑓 (𝑡, 𝑥, 𝑞)]󵄨𝑡=0 = G𝑛,𝑞 [𝑥] . (55) where in the matrix multiplication for every term which 𝑘 𝑘 includes D𝑞𝑓,weoperatewith𝜖 on 𝑔. We denote this by ⋅𝜖. By a routine computation we obtain This operator can also be iterated, compare with1 [ ]. G𝑛,𝑞 [𝑥] G𝑛,𝑞 [𝑦] = G𝑛,𝑞 [𝑥 ⊕𝑞 𝑦] . (56) Theorem 15 (a 𝑞-analogue of [9,page67]). Consider 𝑥⊕ 𝑦 󵄨 The function argument 𝑞 is induced from the corre- P [ (𝑥𝑡)]󵄨 =𝑃 (𝑥) , 𝑞 𝑛,𝑞 E𝑞 󵄨𝑡=0 𝑛,𝑞 (46) sponding -addition. Inthesamewayweobtain where the 𝑞-Pascal matrix is defined in (8). P [⋅] W [⋅] The mappings 𝑛,𝑞 and 𝑛,𝑞 are linear, that is, G𝑛,𝑞 [𝑥] G𝑛,1/𝑞 [𝑦] = G𝑛,𝑞 [𝑥 ⊞𝑞 𝑦] . (57)

P𝑛,𝑞 [𝑎𝑓 (𝑡) +𝑏𝑔(𝑡)]=𝑎P𝑛,𝑞 [𝑓 (𝑡)]+𝑏P𝑛,𝑞 [𝑔 (𝑡)], By the last formula we can define the inverse of G𝑛,𝑞[𝑥] as W𝑛,𝑞 [𝑎𝑓 (𝑡) +𝑏𝑔(𝑡)]=𝑎W𝑛,𝑞 [𝑓 (𝑡)]+𝑏W𝑛,𝑞 [𝑔 (𝑡)]. −1 (47) G𝑛,𝑞[𝑥] ≡ G𝑛,1/𝑞 [−𝑥] . (58) One has the following two matrix multiplication formulas: We now return permanently to [9]. the first one is a 𝑞-analogue of [8,page232], Φ (𝑥) 𝑞 P [𝑓 (𝑡, 𝑞)] ⋅ P [𝑔 (𝑡, 𝑞)] = P [𝑓 (𝑡, 𝑞) 𝑔 (𝑡, 𝑞)]. Definition 18 (compare with [9, page 71]). Let 𝑞 be the - 𝑛,𝑞 𝜖 𝑛,𝑞 𝑛,𝑞 Appell polynomial corresponding to 𝑓(𝑡) in (11). The vector (48) ⃗ of 𝑞-Appell polynomials Φ𝑞(𝑥) for 𝑓(𝑡) is given by P [𝑓 (𝑡, 𝑞)] ⋅ W [𝑔 (𝑡, 𝑞)] = W [𝑓 (𝑡, 𝑞) 𝑔 (𝑡, 𝑞)] . 𝑛,𝑞 𝜖 𝑛,𝑞 𝑛,𝑞 ⃗ Φ𝑞 (𝑥)(𝑖) ≡Φ𝑖,𝑞 (𝑥) , 𝑖=0,1,...,𝑛−1. (59) (49) Theorem 19 𝑞 𝑓(𝑡) The scaling properties can be stated as follows: (a -analogue of [9,page69]).Let be 󵄨 analytic in the neighbourhood of 0,with𝑓(0) = 0.The𝑞- W [𝑓 (𝑎𝑡, 𝑞)]󵄨 𝑛 𝑛,𝑞 󵄨𝑡=0 difference equation in R , 󵄨 (50) = [1,𝑎,...,𝑎𝑛−1] W [𝑓 (𝑡, 𝑞)]󵄨 , 󵄨 diag 𝑛,𝑞 󵄨𝑡=0 ⃗ 󵄨 ⃗ (D𝑞,𝑥) 𝑦 (𝑥) = P𝑛,𝑞 [𝑓 (𝑡, 𝑞)]󵄨𝑡=0 𝑦 (𝑥) , −∞<𝑥<∞, 󵄨 P𝑛,𝑞 [𝑓 (𝑎𝑡, 𝑞)]󵄨𝑡=0 (60) 𝑛−1 󵄨 𝑦(0)⃗ = 𝑦⃗ = diag [1,𝑎,...,𝑎 ] P𝑛,𝑞 [𝑓 (𝑡, 𝑞)]󵄨𝑡=0 (51) with initial value 0, has the following solution: −1 1−𝑛 󵄨 × diag [1, 𝑎 ,...,𝑎 ]. 𝑦⃗ (𝑥) = P [ (𝑓 (𝑡) 𝑥)]󵄨 𝑦⃗ . 𝑛,𝑞 E𝑞 󵄨𝑡=0 0 (61) Proof. The formula (49)isprovedbythe𝑞-Leibniz theorem. Proof. Since E𝑞(𝐴𝑡)|𝑡=0 is equal to the unit matrix, the solu- For (50)usethechainrule. tion of (60)is −1 (𝑘) Definition 16. Assume that 𝑓(0) =0̸ .Iftheinverse(𝑓(𝑡) ) 󵄨 Φ𝑞 (𝑥) = E𝑞 (P𝑛,𝑞 [𝑓 (𝑡)]󵄨 𝑥) 𝑦0⃗ exists for 𝑘<𝑛, we can define the 𝑞-inverse of the generalized 𝑡=0 󵄨 𝑞-Pascal functional matrix as ∞ 󵄨 𝑘 (62) 𝑘 󵄨 𝑥 −1,𝑞 −1 =(∑P𝑛,𝑞 [𝑓 (𝑡)])󵄨 . P [𝑓 (𝑡, 𝑞)] ≡ P [𝑓(𝑡, 𝑞) ]. 󵄨 {𝑘} ! 𝑛,𝑞 𝑛,𝑞 (52) 𝑘=0 󵄨𝑡=0 𝑞 6 Journal of Discrete Mathematics

By (47)and(48) Definition 21 (A 𝑞-analogue of [9,page74]).LetΦ(𝑥)⃗ be the 󵄨 𝑞-Appellpolynomialvectorfor𝑓(𝑡).Thenthe𝑞-Pascal matrix ∞ 𝑥𝑘 󵄨 𝑘 󵄨 for the 𝑞-Appell polynomial vector for 𝑓(𝑡) is defined by Φ𝑞 (𝑥) = P𝑛,𝑞 (∑𝑓 (𝑡) )󵄨 , (63) {𝑘} ! 󵄨 𝑘=0 𝑞 󵄨𝑡=0 󵄨 󵄨 P𝑛,𝑞 [𝑓 (𝑡) E𝑞 (𝑥𝑡)]󵄨 . (67) 𝑘 󵄨𝑡=0 where the power 𝑓 (𝑡) is to be interpreted as 𝑘−1products with ⋅𝜖. Canbecomparedwith(53). ⃗ Theorem 20 (a 𝑞-analogue of [9,page71]).Let Φ(𝑥) be a Theorem 22 (a 𝑞-analogue of [9, pages 74-75]). Let Φ𝜈,𝑞(𝑥), Ξ (𝑥) Ψ (𝑥) 𝑞 vector of length 𝑛.ThenΦ(𝑥)⃗ is the 𝑞-Appell polynomial vector 𝜈,𝑞 ,and 𝜈,𝑞 be the -Appell polynomials corresponding 𝑓(𝑡) 𝑔(𝑡) ℎ(𝑡) for 𝑓(𝑡) if and only if to , ,and ,respectively. 𝑓(𝑡)ℎ(𝑡) = 𝑔(𝑡) 󵄨 󵄨 Further assume that . Φ⃗ (𝑥) = P [𝑓 (𝑡)]󵄨 ⋅ W [ (𝑥𝑡)]󵄨 𝑛,𝑞 󵄨𝑡=0 𝜖 𝑛,𝑞 E𝑞 󵄨𝑡=0 Then 󵄨 (64) 󵄨 𝜈 = W𝑛,𝑞 [𝑓 (𝑡) E𝑞 (𝑥𝑡)]󵄨 . 𝜈 󵄨𝑡=0 Ξ (𝑦 ⊕ 𝑧) = ∑( ) Φ (𝑦) Ψ (𝑧) . 𝜈,𝑞 𝑞 𝑘 𝜈−𝑘,𝑞 𝑘,𝑞 (68) 𝑞 Proof. We know from [2, 17]that 𝑘=0

P𝑛,𝑞 [𝑡]|𝑡=0 =𝐻𝑛,𝑞. (65) Proof. By formula (49)weget 󵄨 We thus obtain P𝑛,𝑞 [𝑓 (𝑡) E𝑞 (𝑦𝑡)]𝜖 ⋅ W𝑛,𝑞 [ℎ (𝑡) E𝑞 (𝑧𝑡)]󵄨 󵄨 󵄨𝑡=0 ⃗ 󵄨 󵄨 Φ (𝑥) = P𝑛,𝑞 [E𝑞 (𝑥𝑡)]󵄨 ⋅𝜖W𝑛,𝑞 [𝑓 (𝑡)]󵄨 󵄨 (69) 󵄨𝑡=0 𝑡=0 = W [𝑓 (𝑡) ℎ (𝑡) ((𝑦 ⊕ 𝑧) 𝑡)]󵄨 . 󵄨 (66) 𝑛,𝑞 E𝑞 𝑞 󵄨𝑡=0 = W [𝑓 (𝑡) (𝑥𝑡)]󵄨 . 𝑛,𝑞 E𝑞 󵄨𝑡=0 By (11) the left-hand side of (69)isequalto

Φ0,𝑞 (𝑦) 0 ⋅ ⋅ ⋅ 0

Φ1,𝑞 (𝑦) Φ0,𝑞 (𝑦) ⋅ ⋅ ⋅ 0 ( ) ( Φ (𝑦) {2} Φ (𝑦) ⋅ ⋅ ⋅ 0 ) ( 2,𝑞 𝑞 1,𝑞 ) ( ) ( . . . . ) . . .. . 𝑛−1 𝑛−1 𝑛−1 ( ) Φ (𝑦) ( ) Φ (𝑦) ⋅ ⋅ ⋅ ( ) Φ (𝑦) 0 𝑛−1,𝑞 1 𝑛−2,𝑞 𝑛−1 0,𝑞 ( 𝑞 𝑞 𝑞 ) (70)

Ψ0,𝑞 (𝑧)

Ψ1,𝑞 (𝑧) ×( ). . .

Ψ𝑛−1,𝑞 (𝑧)

By (11)theright-handsideof(69)isequalto Theorem 23 (another 𝑞-analogue of [9, pages 74-75]). Let Φ𝜈,𝑞 and Ξ𝜈,𝑞 be two 𝑞-Appell polynomial sequences for 𝑓(𝑡) 𝑔(𝑡) Ψ𝑞 𝑞 Ξ0,𝑞 (𝑦𝑞 ⊕ 𝑧) and ,respectively.Let 𝜈 be the pseudo -Appell polynomial for ℎ(𝑡). Furthermore assume that 𝑓(𝑡)ℎ(𝑡) = 𝑔(𝑡).Then Ξ1,𝑞 (𝑦𝑞 ⊕ 𝑧) ( ). (71) . . 𝜈 𝜈 Ξ (𝑦 ⊕ 𝑧) Ξ (𝑦 ⊞ 𝑧) = ∑( ) Φ (𝑦) Ψ𝑞 (𝑧) . 𝑛−1,𝑞 𝑞 𝜈,𝑞 𝑞 𝑘 𝜈−𝑘,𝑞 𝑘 (72) 𝑘=0 𝑞 The theorem follows by equating the last rows of formulas71 ( ) and (70). Proof. By formula (49)weget

󵄨 󵄨 P [𝑓 (𝑡) (𝑦𝑡)]󵄨 W [ℎ (𝑡) (𝑧𝑡)]󵄨 Formula (68) is almost a generalization of (28). 𝑛,𝑞 E𝑞 󵄨𝑡=0 𝑛,𝑞 E1/𝑞 󵄨𝑡=0 There is a variation of this formula with JHC, which (73) enables inverses. = W𝑛,𝑞 [𝑓 (𝑡) ℎ (𝑡) E𝑞 ((𝑦𝑞 ⊞ 𝑧) 𝑡)] |𝑡=0. Journal of Discrete Mathematics 7

By (11)and(13) the left-hand side of (73)isequalto

Φ0,𝑞 (𝑦) 0 ⋅ ⋅ ⋅ 0 Φ1,𝑞 (𝑦) Φ0,𝑞 (𝑦) ⋅ ⋅ ⋅ 0 ( Φ (𝑦) {2} Φ (𝑦) ⋅ ⋅ ⋅ 0 ) ( 2,𝑞 𝑞 1,𝑞 ) ( . . . . ) . . .. . 𝑛−1 𝑛−1 𝑛−1 ( ) Φ (𝑦) ( ) Φ (𝑦) ⋅ ⋅ ⋅ ( ) Φ (𝑦) 0 𝑛−1,𝑞 1 𝑛−2,𝑞 𝑛−1 0,𝑞 ( 𝑞 𝑞 𝑞 ) (74) 𝑞 Ψ0 (𝑧) 𝑞 Ψ1 (𝑧) ×( . ). . 𝑞 Ψ𝑛−1 (𝑧)

By (11)theright-handsideof(73)isequalto Thisisobviouslythesameasthelefthandsideofthe generating function for 𝑞 L-numbers. Ξ0,𝑞 (𝑦𝑞 ⊞ 𝑧) Ξ (𝑦 ⊞ 𝑧) Corollary 26 (a 𝑞-analogue of [9, pages 85-86]). Let Φ𝑛,𝑞(𝑥) 1,𝑞 𝑞 𝑞 ( ). (75) be a 𝑞-Appell polynomial and Ψ (𝑥) apseudo𝑞-Appell polyno- . 𝑛 . mial. Then Ξ (𝑦 ⊞ 𝑧) 𝑛−1,𝑞 𝑞 𝑛 𝑛 𝑛 𝑛 ∑( ) Φ (𝑥) Ψ𝑞 (−𝑥) = ∑( ) Φ Ψ . 𝑘 𝑘,𝑞 𝑛−𝑘 𝑘 𝑘,𝑞 𝑛−𝑘,𝑞 (80) The theorem follows by equating the last rows of formulas 𝑘=0 𝑞 𝑘=0 𝑞 (75)and(74). Let Φ𝑛,𝑞(𝑥) and Ψ𝑛,𝑞(𝑥) be 𝑞-Appell polynomials. Then The following formula from [18,equation(50),page133] 𝑞=1 𝑛 𝑛 𝑛 𝑛 for follows immediately from (68). ∑( ) Φ (𝑦) Ψ (𝑧) = ∑( ) Φ (𝑦⊕ 𝑧) Ψ . 𝑘 𝑘,𝑞 𝑛−𝑘,𝑞 𝑘 𝑘,𝑞 𝑞 𝑛−𝑘,𝑞 𝑠 𝑘=0 𝑞 𝑘=0 𝑞 Corollary 24 ([11,4.242]). When ∑ 𝑛𝑙 =𝑛, 𝑙=1 (81) (𝑛) (𝑥 ⊕ ⋅⋅⋅⊕ 𝑥 ) B𝑁𝑊𝐴,𝑘,𝑞 1 𝑞 𝑞 𝑠 Proof. Formula (80)isprovedasfollows.Weput𝑧=−𝑦in 𝑠 (72). The RHS of (72)isthenequaltotheLHSof(80). We 𝑘 (𝑛 ) (76) 𝑗 finish the proof by using formula (49). Formula (81)isproved = ∑ ( ) ∏B (𝑥𝑗), 𝑚 ,...,𝑚 𝑁𝑊𝐴,𝑚𝑗,𝑞 1 𝑠 𝑞 as follows. Use (68)with𝑓(𝑡) → 𝑓(𝑡)E𝑞(𝑡(𝑦 ⊕𝑞 𝑧)) and ℎ(𝑡). 𝑚1+⋅⋅⋅+𝑚𝑠=𝑘 𝑗=1 Or use an equivalent umbral reasoning. where we assume that 𝑛𝑗 operates on 𝑥𝑗. Corollary 27. One has the following special cases: YangandYounhavefoundanewidentitybetween 𝑛 𝑛 Bernoulli and Euler polynomials: ∑( ) (𝑥) 𝑞 (−𝑥) 𝑘 B𝑁𝑊𝐴,𝑘,𝑞 B𝑁𝑊𝐴,𝑛−𝑘 𝑘=0 𝑞 𝑦+𝑧 𝜈 𝜈 2𝑛 ( ) ∑ ( ) (𝑦) (𝑧) . (82) B𝑛 2 𝑘 B𝑘 E𝑛−𝑘 (77) 𝑛 𝑛 𝑘=0 = ∑( ) , 𝑘 B𝑁𝑊𝐴,𝑘,𝑞B𝑁𝑊𝐴,𝑛−𝑘,𝑞 𝑘=0 𝑞 The 𝑞-analogue looks as follows. 𝑛 𝑛 ∑( ) (𝑦) (𝑧) Corollary 25 (a 𝑞-analogue of [9, page 75 (18)]). Consider 𝑘 B𝑁𝑊𝐴,𝑘,𝑞 B𝑁𝑊𝐴,𝑛−𝑘,𝑞 𝑘=0 𝑞 𝜈 (83) 𝜈 𝑛 L𝑁𝑊𝐴,𝜈,𝑞 (𝑦⊕𝑞 𝑧) = ∑( ) B𝑁𝑊𝐴,𝜈−𝑘,𝑞 (𝑦) F𝑁𝑊𝐴,𝜈,𝑞 (𝑧) . 𝑛 𝑘 = ∑( ) (𝑦 ⊕ 𝑧) , 𝑘=0 𝑞 𝑘 B𝑁𝑊𝐴,𝑘,𝑞 𝑞 B𝑁𝑊𝐴,𝑛−𝑘,𝑞 (78) 𝑘=0 𝑞 𝑛 1 𝑛 Proof. We put in 𝑓(𝑡) =𝑡 /(E𝑞(𝑡)−1) and ℎ(𝑡) = 2/(E𝑞(𝑡)+1) ∑( ) (𝑦) (𝑧) 𝑘 F𝑁𝑊𝐴,𝑘,𝑞 B𝑁𝑊𝐴,𝑛−𝑘,𝑞 in (68). Then 𝑘=0 𝑞 (84) (2𝑡)1 𝑛 𝑛 𝑔 (𝑡) =𝑓(𝑡) 𝑔 (𝑡) = . = ∑( ) (𝑦 ⊕ 𝑧) . 𝑛 (79) 𝑘 F𝑁𝑊𝐴,𝑘,𝑞 𝑞 B𝑁𝑊𝐴,𝑛−𝑘,𝑞 (E𝑞 (𝑡2𝑞)−1) 𝑘=0 𝑞 8 Journal of Discrete Mathematics

Corollary 28 (a 𝑞-analogue of [9,page87]). Consider the 𝑞- Proof. Let Φ𝑛,𝑞(𝑥) be the 𝑞-Appell polynomial for 𝑓(𝑡) and Φ (𝑥) 𝑞 𝑞 Appell polynomial 𝑛,𝑞 and the pseudo- -Appell polynomial Ψ𝑛 (𝑥) the pseudo-𝑞-Appell polynomial for ℎ(𝑡).Byformula 𝑞 Ψ𝑛 (𝑥).Then (50), 𝑛 𝑛 𝑛 𝑛 ∑(−1)𝑘( ) Φ (𝑥) Ψ𝑞 (𝑥) = ∑(−1)𝑘( ) Φ Ψ . 𝑘 𝑘,𝑞 𝑛−𝑘 𝑘 𝑘,𝑞 𝑛−𝑘,𝑞 𝑘=0 𝑞 𝑘=0 𝑞 (85)

󵄨 P [𝑓 (𝑡) (𝑥𝑡)]⋅ W [ℎ (−𝑡) (−𝑥𝑡)]󵄨 𝑛,𝑞 E𝑞 𝜖 𝑛,𝑞 E1/𝑞 󵄨𝑡=0

Φ0,𝑞 (𝑥) 0⋅⋅⋅0 Φ (𝑥) Φ (𝑥) ⋅⋅⋅ 0 𝑞 1,𝑞 0,𝑞 Ψ0 (𝑥) Φ (𝑥){2} Φ (𝑥) ⋅⋅⋅ 0 −Ψ𝑞 𝑥 (86) ( 2,𝑞 𝑞 1,𝑞 ) 1 ( ) = ( . . . . ) ( . )...... 𝑛−1 𝑛−1 𝑛−1 (−1)𝑛+1Ψ𝑞 (𝑥) ( ) Φ (𝑥) ( ) Φ (𝑥) ⋅⋅⋅ ( ) Φ (𝑥) 𝑛−1 0 𝑛−1,𝑞 1 𝑛−2,𝑞 𝑛−1 0,𝑞 ( 𝑞 𝑞 𝑞 )

On the other hand, by formula (49), we obtain

󵄨 P [𝑓 (𝑡) (𝑥𝑡)]⋅ W [ℎ (−𝑡) (−𝑥𝑡)]󵄨 = P [𝑓 (𝑡)]⋅ W [ℎ (−𝑡)]| 𝑛,𝑞 E𝑞 𝜖 𝑛,𝑞 E1/𝑞 󵄨𝑡=0 𝑛,𝑞 𝜖 𝑛,𝑞 𝑡=0

Φ0,𝑞 (0) 0⋅⋅⋅0 Φ (0) Φ (0) ⋅⋅⋅ 0 𝑞 1,𝑞 0,𝑞 Ψ0 (0) Φ (0){2} Φ (0) ⋅⋅⋅ 0 −Ψ𝑞 0 (87) ( 2,𝑞 𝑞 1,𝑞 ) 1 ( ) = ( . . . . ) ( . )...... 𝑛−1 𝑛−1 𝑛−1 (−1)𝑛+1Ψ𝑞 (0) ( ) Φ (0) ( ) Φ (0) ⋅⋅⋅ ( ) Φ (0) 𝑛−1 0 𝑛−1,𝑞 1 𝑛−2,𝑞 𝑛−1 0,𝑞 ( 𝑞 𝑞 𝑞 )

The theorem follows by multiplying both sides of (85)by Proof. By formula (49)weget 𝑛 (−1) and equating the last rows of formulas (86)and 󵄨 𝑔 (𝑡) 󵄨 (87). Φ⃗ (𝑥) = W [𝑓 (𝑡) (𝑥𝑡)]󵄨 𝑞 𝑛,𝑞 E𝑞 󵄨 𝑔 (𝑡) 󵄨𝑡=0 (90) 𝑓 (𝑡) 󵄨 We can easily write down specializations of (85)to = P [ ]⋅ W [𝑔 (𝑡) (𝑥𝑡)]󵄨 . 𝑛,𝑞 𝑔 (𝑡) 𝜖 𝑛,𝑞 E𝑞 󵄨𝑡=0 (pseudo) 𝑞-Bernoulli and 𝑞-Euler polynomials. One example is Corollary 30 (a 𝑞-analogue of [9,page77]). Arelationship 𝑛 𝑛 ∑(−1)𝑘( ) (𝑥) 𝑞 (𝑥) between Bernoulli and Euler polynomials 𝑘 BNWA,𝑘,𝑞 BNWA,𝑛−𝑘 𝑞 𝑘=0 {𝑛} (𝑥) (88) 𝑞F𝑁𝑊𝐴,𝑛−1,𝑞 𝑛 B𝑁𝑊𝐴,𝑛,𝑞 (𝑥) = 𝑘 𝑛 2 = ∑(−1) ( ) B ,𝑘,𝑞B ,𝑛−𝑘,𝑞. 𝑘 NWA NWA (91) 𝑘=0 𝑞 𝑛 𝑛 + ∑( ) (𝑥) . 𝑘 B𝑁𝑊𝐴,𝑛−𝑘,𝑞F𝑁𝑊𝐴,𝑘,𝑞 𝑞 Theorem 29 (the 𝑞-connection constant theorem, a 𝑞-anal- 𝑘=0 ⃗ ⃗ ogue of [9,page77]).Let Φ𝑞(𝑥) and Ψ𝑞(𝑥) be the 𝑞-Appell 1 Proof. We put in 𝑓(𝑡) =𝑡 /(E𝑞(𝑡)−1) and 𝑔(𝑡) = 2/(E𝑞(𝑡)+1) 𝑓(𝑡) 𝑔(𝑡) vectors corresponding to and ,respectively.Then in (89). Then 󵄨 1 1 𝑓 (𝑡) 󵄨 𝑓 (𝑡) 𝑡 𝑡 Φ⃗ (𝑥) = P [ ]󵄨 Ψ⃗ (𝑥) . = + . 𝑞 𝑛,𝑞 󵄨 𝑞 (89) 𝑔 (𝑡) 2 (𝑡) −1 (92) 𝑔 (𝑡) 󵄨𝑡=0 E𝑞 Journal of Discrete Mathematics 9

We find that Theorem 31 (a 𝑞-analogue of [9,page91]). Let Φ𝑛,𝑞(𝑥) be the 𝑞-Appell polynomial sequence for 𝑓(𝑡).ThenΦ𝑛,𝑞(𝑥) satisfies the following 𝑞-difference equation:

𝑛 1 𝑛 𝛼𝑘−1 𝑘 −1 { + , 𝑘=1, ∑𝑞 D𝑞𝑓(𝑥𝑞 ) (𝑘) { BNWA,𝑘,𝑞 if {𝑘−1} ! 𝑓 (𝑡) {2 𝑘=2 𝑞 ( ) (0) = 𝑔 (𝑡) { (93) { +𝑞𝑛 (𝑥𝑞 +𝛼 ) 𝑓(𝑥𝑞−1) (94) . 0 D𝑞 {BNWA,𝑘,𝑞 otherwise 𝑛 −1 − {𝑛+1}𝑞𝑓 (𝑥) +𝑞 𝑓(𝑥𝑞 )=0,

𝑘 where 𝛼𝑘 ≡ D𝑞(−𝑓(𝑡)D𝑞(1/𝑓(𝑡)))|𝑡=0. The result follows by (89). Proof. We put 𝑓(𝑡) = 1/𝑔(𝑡):

Φ1,𝑞 (𝑥) Φ (𝑥) 𝑥𝑡 󵄨 2,𝑞 E𝑞 ( ) 󵄨 ( )=W [ ( )]󵄨 . 𝑛,𝑞 D𝑞 𝑔 (𝑡) 󵄨 . 󵄨𝑡=0 Φ𝑛,𝑞 (𝑥) 𝑔 (𝑡) (𝑥𝑡) 󵄨 (𝑥𝑡) 𝑔 (𝑡) 󵄨 D𝑞 E𝑞 󵄨 by (50) E𝑞 D𝑞 󵄨 = W [(𝑥 − ) ]󵄨 = P [ ]⋅ W [𝑥 − ]󵄨 𝑛,𝑞 𝑔 𝑡 󵄨 𝑛,𝑞 𝜖 𝑛,𝑞 𝑔 𝑡 󵄨 ( ) 𝑔(𝑞𝑡) 󵄨𝑡=0 𝑔(𝑞𝑡) ( ) 󵄨𝑡=0 (95) −1 Φ0,𝑞 (𝑥𝑞 ) 0 ⋅⋅⋅ 0 −1 −1 𝑞Φ1,𝑞 (𝑥𝑞 )Φ0,𝑞 (𝑥𝑞 ) ⋅⋅⋅ 0 𝑥+𝛼0 2 −1 −1 ( 𝑞 Φ (𝑥𝑞 )𝑞{2} Φ (𝑥𝑞 ) ⋅⋅⋅ 0 ) 𝛼1 = ( 2,𝑞 𝑞 1,𝑞 ) ( ). ( . . . . ) ...... 𝑛−1 𝑛−1 𝑛−1 𝛼𝑛−1 𝑞𝑛−1( ) Φ (𝑥𝑞−1)𝑞𝑛−2( ) Φ (𝑥𝑞−1) ⋅⋅⋅ ( ) Φ (𝑥𝑞−1) 0 𝑛−1,𝑞 1 𝑛−2,𝑞 𝑛−1 0,𝑞 ( 𝑞 𝑞 𝑞 )

By equating the first and last expression for the last rows of where we have used the following identities: formula (95)weget(𝜈=0,1,...,𝑛) 𝑘 −1 −𝑘 D𝑞Φ𝑛,𝑞 (𝑥𝑞 )=𝑞 {𝑛−𝑘+1}𝑘,𝑞Φ𝑛−𝑘,𝑞, 𝜈−1 −1 (98) Φ𝜈,𝑞 (𝑥) =(𝑥+𝛼0)𝑞 Φ𝜈−1,𝑞 (𝑥𝑞 ) 𝑛 −1 D𝑞Φ𝑛−1,𝑞 (𝑥𝑞 )=0. 𝜈−1 𝜈−1 (96) + ∑( ) 𝛼 𝑞𝜈−𝑘−1Φ (𝑥𝑞−1). 𝑘 𝑘 𝜈−𝑘−1,𝑞 𝑞 𝑘=1 4. Conclusion 𝑞 𝜈−1=𝑛 By finding -analogues of Yang and Youn [9], we have also Operating with D𝑞 on this and putting , shown that Yang-Youn paper contains a lot of new and inter- esting formulas. It is our hope that this paper together with future articles will point out the interplay between special {𝑛+1} Φ (𝑥) 𝑞 𝑛,𝑞 functions and linear algebra. The formulas in Corollary 28 areinthesamestyleasNørlund’s[18]formulas,whichare =𝑞𝑛Φ (𝑥𝑞−1) 𝑛,𝑞 proved using umbral calculus. However, our formulas (78), 𝑛−1 −1 Corollary 30,and(91)areofamoreintricatenature.This +𝑞 {𝑛}𝑞 (𝑞𝑥+𝛼0)Φ𝑛−1,𝑞 (𝑥𝑞 ) (97) shows that the matrix approach in this paper is not just a restatement of Nørlund’s [18]umbralcalculusin𝑞-deformed 𝑛 𝑛 + ∑( ) 𝛼 𝑞𝑛−𝑘−1{𝑛−𝑘} form but a generalization. We also see that in order to obtain 𝑘 𝑘 𝑞 𝑘=1 𝑞 𝑞-analogues of the Yang and Youn [9,pages85-87]formulas, we had to use (pseudo) 𝑞-Bernoulli and 𝑞-Euler polynomials, −1 ×Φ𝑛−𝑘−1,𝑞 (𝑥𝑞 ), 𝑛=0,1,...,𝑛, which were first mentioned in [5]. 10 Journal of Discrete Mathematics

Appendix [14] E. D. Rainville, Special Functions,Chelsea,Bronx,NY,USA,1st edition, 1971, Reprint of 1960 first edition. Bernoulli Number Formulas in Hansen [15]P.Appell,“Suruneclassedepolynomes,”ˆ Annales Scientifiques de l’Ecole´ Normale Superieure´ ,vol.9,pp.119–144,1880. In Hansens impressive table [10,pages331–347],several [16] Y. Yang, “Generalized Leibniz functional matrices and factor- formulas for Bernoulli and Euler numbers are listed. Weshow izations of some well-known matrices,” Linear Algebra and Its 𝑞=1 that some of these are connected to our formulas. For , Applications,vol.430,no.1,pp.511–531,2009. formula (83) can be written as follows: [17] T. Ernst, “An umbral approach to find 𝑞-analogues of matrix formulas,” Submitted. 𝑛 𝑛 ∑ ( ) (𝑦) (𝑧) 𝑘 B𝑘 B𝑛−𝑘 [18] N. E. Nørlund, Differenzenrechnung, Springer, 1924. 𝑘=0 (A.1) 𝑛 𝑛 = ∑ ( ) (𝑦 + 𝑧) . 𝑘 B𝑘 B𝑛−𝑘 𝑘=0

Hansen [10, page 341, 50.11.2] writes this as follows:

𝑛 (−𝑛) ∑(−1)𝑘 𝑘 (𝑦) (𝑧) 𝑘! B𝑘 B𝑛−𝑘 𝑘=0

=𝑛(𝑦+𝑧−1)B𝑛−1 (𝑦 + 𝑧) − (𝑛−1) B𝑛 (𝑦 + 𝑧) . (A.2)

Hansen [10,page346,51.6.2]alsohasacorrespondingresult fortheEulerpolynomials.

References

[1] T. Ernst, “𝑞-Leibniz functional matrices with connections to 𝑞- Pascal and 𝑞-Stirling matrices,” Advanced Studies in Contempo- rary Mathematics,vol.22,no.4,2012. [2] T. Ernst, “An umbral approach for 𝑞-Pascal matrices,” Submit- ted. [3] L. Carlitz, “Products of Appell polynomials,” Collectanea Math- ematica,vol.15,pp.245–258,1963. [4] P. Nalli, “Sopra un procedimento di calcolo analogo alla inte- grazione,” Rendiconti del Circolo Matematico di Palermo,vol.47, no. 3, pp. 337–374, 1923. [5] T. Ernst, “𝑞-Bernoulli and 𝑞-Euler polynomials, an umbral approach II,” U.U.D.M. Report 2009:17, 2009. [6] W. A. Al-Salam, “𝑞-Appell polynomials,” Annali di Matematica Pura ed Applicata,vol.77,no.1,pp.31–45,1967. [7] T. Ernst, “𝑞-Bernoulli and 𝑞-Euler polynomials, an umbral approach,” International Journal of Difference Equations,vol.1, no. 1, pp. 31–80, 2006. [8] Y. Yang and C. Micek, “Generalized Pascal functional matrix and its applications,” Linear Algebra and Its Applications,vol. 423, no. 2-3, pp. 230–245, 2007. [9] Y. Yang and H. Youn, “Appell polynomial sequences: a linear algebra approach,” JPJournalofAlgebra,NumberTheoryand Applications,vol.13,no.1,pp.65–98,2009. [10] E. Hansen, ATableofSeriesandProducts, Prentice Hall, 1975. [11] T. Ernst, A Comprehensive Treatment of q-Calculus,Birkhauser,¨ 2012. [12] W. Hahn, Lineare geometrische Differenzengleichungen,vol.169, Forschungszentrum Graz Mathematisch-Statistische Sektion, Graz, Austria, 1981. [13] L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, UK, 1951. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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