International Journal of Latest Research in Engineering and Technology (IJLRET) ISSN: 2454-5031 www.ijlret.comǁ Volume 2 Issue 3ǁ March 2016 ǁ PP 01-13 www.ijlret.com 1 | Page Some more q -Methods and their applications Prashant Singh 1 , Pramod Kumar Mishra 2 1 (Department of Computer Science, Institute of Science, Banaras Hindu University, India) 2 (Department of Computer Science, Institute of Science, Banaras Hindu University, India) Abstract : This paper is a collection of q analogue of various problems. It also aims at focusing on work performed by various researchers and describes q analogues of various functions. We have also proposed q analogue of some integral transforms (viz. Wavelet Transforms, Gabor Transform etc.) Keywords -q analogue, basic analogue, q method, classical method, basic hyper-geometric function I. INTRODUCTION AND LITERATURE SURVEY C.F.Gauss [1, 11] started the theory of q hyper-geometric series in 1812 and worked on it for more than five decades and he presented the series 1+    +  +1( +1) 1.2. ( +1)  2 + ⋯ (1.1) , where a, b, c and z are complex numbers and c = 0, −1, −2, ...,at the Royal Society of Sciences, Gottingen. Thirty three years later E. Heine [1,11] converted a simple observation lim →ݍ1 1− ݍ  1− ݍ =  (1.2) into a systematic theory of basic hyper-geometric series (q-hyper-geometric series or q-series) 1+ (1− ݍ  ) (1− ݍ) (1− ݍ  ) (1− ݍ  )  + (1.3) In fact, the theory was started in 1748, when Euler [1, 11] considered the infinite product  (1 − ݍ  ) −1 ∞  =1 (1.4) as a generating function for p(n), the number of partitions of a positive integer n, partition of a positive integer n is being a finite non-increasing sequence of positive integers whose sum is n. During 1860−1890, some more contributions to the theory of basic hyper-geometric series were made by J. Thomae and L. J. Rogers. In the beginning of twentieth century F. H. Jackson [1,11,17,18,19,60] started the program of developing the theory of basic hyper-geometric series in a systematic manner, studying q-differentiation, q-integration and deriving q- analogues of the hyper-geometric summation and transformation formulae that were discovered by A. C. Dixon, J. Dougall [1],L. Saalsch¨utz, F. J. W. Whipple[1] and others. During the same time Srinivasa Ramanujan has also made significant contributions to the theory of hyper-geometric and basic hyper-geometric series by recording many identities involving hyper-geometric and basic hyper-geometric series in his notebooks, which were later brought before the mathematical world by G. H. Hardy. During 1930’s and 1940’s many important results on hyper-geometric and basic hyper-geometric series were derived by W. N. Bailey[1]. Of these Bailey’s transform is considered as Bailey’s greatest work. The main contributors to the theory during 1950’s are D. B. Sears, L. Carlitz, W. Hahn [1,11] and L. J. Slater [1,11]. In fact, Sears [1,11]derived several transformation formulae for 3φ2-series, balanced 4φ3-series and very-well-poised r+1 φ r -series. After 1950, the theory of hyper- geometric and basic hyper-geometric series becomes an active field of research, kudos to R.P.Agrawal [53,54,55,56,57], G. E. Andrews [1,11,51,52] and R. Askey[11]. F.H.Jackson [1,11,17,18,19] proposed q-differentiation and q-integration and worked on transformation of q- series and generalized function of Legendre and Bessel. G.E.Andrews [11,51,52] contributed a lot on q theory and worked on q-mock theta function, problems and prospects on basic hyper-geometric series, q-analogue of Kummer’s Theorem. G.E.Andrew [11,51,52] with R.Askey [1] worked on q extension of Beta Function. J.Dougall [1] worked on Vondermonde’s Theorem. H.Exton [1] worked a lot on basic hyper-geometric function and its applications.