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On Q-Integral Representations for Q-Hahn and Askey-Wilson Polynomials and Certain Generating Functions∗

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On Q-Integral Representations for Q-Hahn and Askey-Wilson Polynomials and Certain Generating Functions∗ Applied Mathematics E-Notes, 17(2017), 186-198 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ On q-Integral Representations For q-Hahn And Askey-Wilson Polynomials And Certain Generating Functions Jian Caoy, Zhong-Hua Shenz Received 1 August 2016 Abstract In this paper, q-integral representations for q-Hahn and Askey-Wilson poly- nomials are given by the method of q-difference equation. Moreover, Srivastava- Agarwal type generating function for Al-Salam-Chihara polynomials and mixed generating function for Andrews-Askey polynomials are deduced by means of q-integral. At last, duality property of q-Hahn polynomials is obtained by the technique of transformation. 1 Introduction The q-polynomials and their generating functions are very important and interesting sets of special functions and more specially of orthogonal polynomials. They appear in several branches of mathematics [2, 16, 29], e.g., continued fractions, Eulerian series, elliptic functions, quantum groups and algebras, discrete mathematics (combinatorics, graph theory), coding theory, etc. For more information, we refer to [1, 2, 15, 16, 24, 29, 32, 33, 34, 35, 38]. In this paper, we follow the notations and terminology in [16] and suppose that 0 < q < 1. The q-series and its compact factorials are defined respectively by n 1 k 1 k (a; q)0 = 1, (a; q)n = 1 aq , (a; q) = 1 aq 1 k=0 k=0 Y Y and (a1, a2, . , am; q)n = (a1; q)n(a2; q)n (am; q)n, where m is a positive integer and n is a non-negative integer or . ··· 1 The basic hypergeometric series rs [16, Eq. (1.2.22)] is given by s+1 r a1, . , ar 1 (a1, a2, . , ar; q)n n n n(n 1)/2 rs ; q, z = z ( 1) q , b1, . , bs (q, b , . , b ; q) n=0 1 s n X h i Mathematics Subject Classifications: 05A30, 11B65, 33D15, 33D45. yDepartment of Mathematics, Hangzhou Normal University, Hangzhou City, Zhejiang Province, 310036, P.R. China zDepartment of Mathematics, Hangzhou Normal University, Hangzhou City, Zhejiang Province, 310036, P.R. China 186 J. Cao and Z.-H. Shen 187 which is convergent for either q < 1 and z < when r s or q < 1 and z < 1 when r = s + 1, provided thatj noj zero appearsj j in1 the denominator.≤ j j j j The continuous q-Hermite polynomials [23, Eq. (3.26.1)] n in q , 0 n 2i Hn(x q) = e 2 ; q, q e , x = cos (1) j 0 and the big q-Hermite polynomials [23, Eq. (3.18.1)] n i i 1 q , ae , ae Hn(x; a q) = 3 ; q, q , x = cos . (2) j an 2 0, 0 The Al-Salam-Chihara polynomials [23, Eq. (3.8.1)] n i i (ab; q)n q , ae , ae Qn(x; a, b q) = 3 ; q, q , x = cos . j an 2 ab, 0 The continuous dual q-Hahn polynomials [23, Eq. (3.3.1)] n i i (ab, ac; q)n q , ae , ae pn(x; a, b, c q) = 3 ; q, q , x = cos . (3) j an 2 ab, ac Those q-polynomials are based on the Askey-scheme [23]. For more information, we refer to [16, 23]. The Askey-Wilson polynomials [23, Eq. (3.1.1)] (ab, ac, ad; q) q n, abcdqn 1, aei, ae i p (a, b, c, d; cos ) = n ; q, q n an 4 3 ab, ac, ad are the 43 polynomials. Note that the product of the denominator parameters in this 43 series is q times the product of the numerator parameters, and the argument of the function is q. Basic hypergeometric series with this property are called balanced [16]. The Askey-Wilson polynomials are the most general family of orthogonal polynomials that share the properties of the classical polynomials of Jacobi, Hermite and Laguerre. The Askey-Wilson polynomials provide a basic generalization or a q-analog of Wigner’s 6 j symbols and the Racah coeffi cients. For more information, please refer to [4, 18, 19, 20, 21, 26, 27, 36]. The concept of the q-integral has proved to be very useful in analyzing q-special functions. Ismail and Stanton [20] deduced the following q-integral representations for q-Hermite and Al-Salam-Chihara polynomials [20, Theorem 2.1 and 4.1] i 1 e H(cos q) = yn qyei, qye i; q d y, i 2i 2i q j (1 q)e q, qe , e ; q i Ze 1 1 n i i i i ei i i t t1e , t1e , t2e , t2e ; q qye , qye ; q Q (cos ; t , t q) = 1 yn d y. n 1 2 i 2i 2i 1 1 q j (1 q)e q, t t , qe , e ; q i (t1y, t2y; q) 1 2 e 1 Z 1 188 On q-integral Representations The author [9] given the q-integral representations for Andrews-Askey polynomials [9, Eq. (40)] n (a,b) n (ay, by; q)k k n k An (x, y q) , x y j k (abxy; q)k k=0 X (ax, ay, bx, by; q) y tn qt/x, qt/y; q = 1 1 dqt. (4) y(1 q) q, qy/x, x/y, abxy; q x (at, bt; q) 1 Z 1 Motivated by [20], one may ask naturally a question: Whether there exist q-integral representations for q-Hahn and Askey-Wilson polynomials? For more information, please refer to [7, 9, 10, 20, 30, 31, 37]. The purpose of this paper is to answer the above question and represent Askey- Wilson polynomials in terms of q-integrals as follows. THEOREM 1. For n N, we have 2 q n, ax, ay, acdxy /q ; q, q 4 3 abxy, acxy, adxy an ax, ay, bx, by, acdxy /q, cy, dy; q y tn qt/x, qt/y; q = 1 1 y(1 q)q, qy/x, x/y, abxy, acxy, adxy, cdy ; q x (at, bt; q) 1 Z 1 x/t, q/(by); q (cy)j j 1 j q , q/(cy), q/(c ) 3 ; q, q dqt. (5) 8 q, q/(bt); q 2 q2/(cdy ), 0 9 j=0 j <X = : ; COROLLARY 2. For n N, we have 2 pn(a, b, c, d; cos ) i i i i n 1 i i ei n i i ae , ae , be , be , abcdq , ce , de ; q t qte , qte ; q = i 2i 2i n n n n i1 1 e (1 q) q, qe , e , abq , acq , adq , bcdq e ; q i (at, bt; q) e 1 Z 1 e i/t, qe i/b; q (cei)j j i 1 n 1 j q , qe /c, q /(bc) 32 2 n i ; q, q dqt. (6) 8 q, q/(bt); q q e /(bcd), 0 9 j=0 j <X = : ; COROLLARY 3. For n N, we have 2 pn(x; a, b, c q) i j i i i i ae , ae , be , be , ce ; q = 1 ein(1 q) q, qe2i, e 2i, abqn, acqn; q i 1 e n i i i i t qte , qte ; q e /t, qe /b i 1 21 ; q, ce dqt. (7) e i (at, bt; q) q/(bt) Z 1 J. Cao and Z.-H. Shen 189 i i n i i n REMARK 4. Letting (x, y, ) = e , e , bq and (x, y, d, ) = e , e , 0, bq in Theorem 1, equation (5) reduces to (6) and (7) respectively. The rest of the paper will be organized as follows. In section 2, we deduce the main theorem by the method of q-difference equation. In section 3, we gain a new Srivastava-Agarwal type generating function for Al-Salam-Chihara polynomials by us- ing q-integral. In section 4, we obtain duality property of q-Hahn polynomials by technique of transformation. In section 5, we derive a mixed generating function for Andrews-Askey polynomials by means of q-integral. 2 Proof of the Main Theorem In this paper, we follow the notations and terminology in [16] and suppose that 0 < q < 1. The q-series and its compact factorials are defined respectively by n 1 k 1 k (a; q)0 = 1, (a; q)n = 1 aq , (a; q) = 1 aq 1 k=0 k=0 Y Y and (a1, a2, . , am; q)n = (a1; q)n(a2; q)n (am; q)n, where m is a positive integer and n is a non-negative integer or . ··· Jackson defined the q-integral1 by [22] d d c 1 1 n n f(t)dqt = d(1 q) f dq q , f(t)dqt = f(t)dqt f(t)dqt. Z0 n=0 Zc Z0 Z0 X For more information about q-series and q-integrals, please refer to [9, 16]. The follow- ing two q-difference operators are defined by [14, 16] 1 f(a) f(aq) f aq f(a) D f(a) = , f(a) = . a a a q 1a f g f g Chen and Liu [13, 14] employed the technique of Parameter Augmentation by con- structed the following two q-exponential operators n (n) n 1 bDa 1 q 2 ba (bD ) = , (b ) = . T a (q; q) E a (q; q) n=0 n n=0 n X X Later, authors [12, 15] researched on the following general q-exponential operators 1 (a; q) 1 (a; q) (a, bD ) = n (bD )n, (a, b ) = n ( b )n. T c (q; q) c E c (q; q) c n=0 n n=0 n X X To prove Theorem 1, the following lemmas are necessary.
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