Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Parameter Augmentation for Two Formulas. | Parameter Augmentation for Two Formulas Caihuan Zhang Department of Mathematics DaLian University of Technology Dalian 116024 P. R. China zhcaihuan@ Submitted Jun 5 2006 Accepted Nov 7 2006 Published Nov 17 2006 Mathematics Subject Classifications 33D15 05A30 Abstract In this paper by using the q-exponential operator technique on the q-integral form of the Sears transformation formula and a Gasper q-integral formula we obtain their generalizations. 1 Notation In this paper we follow the notation and terminology in 4 . For a real or complex number q q 1 . let 1 A 1 A q i JJ 1 - -q n 1 and let A q M be defined by Afe for arbitrary parameters A and ụ so that J1 n 0 A n A q n t 1-A 1-Aq . 1-Aqn-l neN 1 2 3 - The q-binomial coefficient is defined by n 1 q n -fc-l q k q n-k Further recall the definition of basic hypergeometric series sộs 1 1 s @11 1 @s 1 q z 1 s n n n 0 q. 11 03-1 n THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N19 1 Here we will frequently use the Cauchy identity and its special case 4 ax q i X a q nXn x q i n 0 q q n 1 _1 n x x q i n 0 q -. -x q i X q n xn t 0 q 2 The exponential operator T bDq The usual q-differential operator or q-derivative is defined by Dq if a f a -af aq By convention D0 is understood as the identity. The Leibniz rule for Dq is the following identity which is a variation of the q-binomial theorem 1 Dff a g a Ê qk k- k 0 In 3 Chen and Liu construct a q-exponential operator based on this denoted T TD X n 0 q n For T bdq there hold the following operator identities. 1 at bt q i abst q i n Dkif a D - a T bDq at q i T bDq as at q i as at bs bt q 1 3 A generalization of the q-integral form of the sears transformation In this section we consider the following formula 3 Theorem rd qt c qt d abcdet q 1 d 1 q q dq c c d abcd bcde acde q 1 ------- . . X-------dqt Jc at bt et q i ac ad bc bd ce de q 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N19 2 Chen and Liu showed it can be .
