IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 5 Ver. I (Sep. - Oct. 2015), PP 95-101 www.iosrjournals.org DOI: 10.9790/5728-115195101 www.iosrjournals.org 95 | Page Infinite Series Obtained By Backward Alpha Difference Operator With Real Variable M.Maria Susai Manuel 1 , G.Dominic Babu 2 , G.M.Ashlin 3 and G.Britto Antony Xavier 4 1 Department of Mathematics, R.M.D. Engineering College, Kavaraipettai - 601 206, Tamil Nadu, S.India. 2,3 Department of Mathematics, Annai Velankanni College, Tholaiyavattam, Kanyakumari District, Tamil Nadu, S.India. 4 Department of Mathematics, Sacred Heart College, Tirupattur, Vellore District. Abstract: In this paper, we derive the formula for Infinite Multi-Series generated by generalized backward alpha difference equation by equating the closed and infinite summation form solutions of higher order generalized i   backward difference equation for positive and negative variable. Suitable examples are inserted to illustrate the main results. Mathematics Subject Classification: 39A70, 47B39, 39A10. Keywords: Generalized alpha difference equation, infinite multi-alpha series, Closed form solution. I. Introduction The modern theory of differential or integral calculus began in the 17 th century with the works of Newton and Leibnitz. In 1989, K.S.Miller and Ross [11] introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the fractional derivative operator. In 2011, M.Maria Susai Manuel, et.al, [8, 11] extended the definition of   to ()    defined on () uk as () ( )= ( ) () vk vk vk        , where 0   , >0  are fixed and [0, ) k   is variable. The results derived in [11] are coincide with the results in [7] when =1  . An equation involving both  and   is called mixed difference equation. Oscillatory behaviour of solutions certain types of mixed difference equations have been dicussed in [3, 4, 6, 12]. In 2014, G.Britto Antony Xavier, et al. [2], [3] proved several interesting results of geometric progression using generalized difference operator and q-difference operator. In this paper, we obtain infinite summation form and closed form solution of higher order backward   difference equation for getting formula of infinite multi-series of polynomials. II. Preliminaries In this section, we define the generalized backward alpha difference operator and we presents certain results on its inverse alpha difference operator with polynomial and polynomial factorials for positive and negative variable k . Definition 2.1 If () vk is a real valued function on [0, )  , then the generalized difference operator for negative  denoted by ( )     is defined as ( ) ( )= ( ) ( ), (0, ) vk vk vk            (1) If ( ) ( )= () vk uk     then the inverse generalized   difference equation is defined as 1 ( ) ( )= () vk uk      Definition 2.2 The higher order generalized i   difference equation is defined as ( ) ( ) 1 1 2 2 ( )= ( ), [0, ), >0 i uk k             (2)