Int. J. Pure Appl. Sci. Technol., 15(2) (2013), pp. 68-72 International Journal of Pure and Applied Sciences and Technology ISSN 2229-6107 Available online at www.ijopaasat.in Research Paper Generalization of Summation Theorems Due to Ramanujan and their Uses in Finding Some Curious Results S. Ismail Azad 1, *, Intazar Husain 2 and Mohd. Saeed Akhtar 1 1 General Studies Department, Yanbu Industrial College, Yanbu, P.O. Box - 30436, KSA - 51000 2 Department of Applied Sciences & Humanities, Delhi Institute of Technology Management and Research, Faridabad, Haryana, India - 121004 * Corresponding author, e-mail: (dr.ismailazad@gmail.com) (Received: 1-3-13; Accepted: 2-4-13) Abstract: The aim of this paper to provide the simple way for the generalization of three summation due to Ramanujan and then to use them to find some curious results by applying the theory of hypergeometric function. The result is derived with the help of generalized Dixon theorem available in the literature. The results derived in this research note are simple, interesting, easily established and may be potentially useful. Keywords: Generalized Hypergeometric Function; Dixon s Summation Theorem; Generalized Dixon s Summation Theorem, Ramanujan s Summations. 1. Introduction In the theory of hypergeometric and generalized hypergeometric series, classical summations such as of Gauss, Gauss second, Kummer and Bailey for the series 2 F 1 ; Watson, Dixon, Whipple and Saalschütz for the series 3 F 2 ; generalized Dixon s summation theorem for the series 5 F 4 and others play an important role [1], [2], [3], [6]. Applications of the above mentioned classical summation theorems are now well known. We start with the following interesting summations due to Ramanujan [4]. 1+ + + = (1.1)
Int. J. Pure Appl. Sci. Technol., 15(2) (2013), 68-72 69 1+ + + = And another infinite sum due to Ramanujan [1], [3] is (8+1) Γ(+0.25)!Γ(0.25) = (1.2) =1+9 +17 +25 + = (.) (1.3) As pointed out by Berndt [4] that the above summation (1.1) and (1.2) due to Ramanujan can be obtained quite simply by choosing suitable parameters, and and then applying following classical Dixon theorem [3],, ;1 1+,1+ = Γ1+ 2 Γ(1+ ) Γ(1+ ) Γ1+ 2 Γ(1+) Γ1+ 2 Γ1+ 2 Γ(1+ ) Provided,R( 2 2)> 2 (1.4) The aim of this research note is to provide another natural generalization of the Ramanujan s summation (1.1) (1.2) and (1.3) and also some Ramanujan type series [5]. For this, the following generalized Dixon s summation theorem, generalized hypergeometric summation theorems, few recurrence formulae and identities will be required in our present investigations. Generalized Dixon s Summation Theorem, +1,,, 2 ; 1 2, 1+, 1+, 1+ = Γ(1+ ) Γ(1+ ) Γ(1+ ) Γ(1+ ) Γ(1+) Γ(1+ ) Γ(1+ ) Γ(1+ ) Provided, Re ( ) > 1 (1.5) Generalized Hypergeometric Summation Theorem,;: 1=1+ 1! + (+1)(+1) + (+1)(+2)(+1)(+2) + 2!(+1) 3!(+1)(+2) = () () ()! (1.6) which converges if is not a negative integer (1) for all of <1 and (2) on the unit circle =1 if ( )>0. Here, () is the Pochhammer symbol or rising factorial.
Int. J. Pure Appl. Sci. Technol., 15(2) (2013), 68-72 70 Pochhammer Symbol The Pochhammer symbol or shifted factorial is defined by () = Γ(+) Γ() 1 ; =0 = (+1)(+2) (+ 1) ; =1,2,.. Note that (0) =1. 0, 1, 2,, and the notation Γ stands for Gamma function. Double Factorial The double factorial of a positive integer is a generalization of the usual factorial! defined by ( 2) 5 3 1 >0 odd!!= ( 2) 6 4 2 >0 even 1 = 1,0 Note that 1!!=0!!=1, by definition (1, Arfken 1985, p. 547). Recurrence Relations and Identities Γ() Γ( )= sin() Γ() Γ(1 )= sin() Γ(1+)= Γ() Γ(1 )= Γ( ) Γ()=( 1)! + += Several of these are also given in [1], [2]. 2. Main Results The following generalized summation will be established in this research note. Case 1: For <1: 1+ 1 5 (3 2) +1 6 (+1) (3 2)(5 2) + 5 26 (+1)(+2) (3 2)(5 2)(7 2) + Case 2: For <0.75: =. Γ(1 )Γ(1.5 ) 4Γ(0.75) (2.1) Γ(1.25 ) 1+ 1 2 (3 2) + 9 16 (+1) (3 2)(5 2) +25 32 (+1)(+2) (3 2)(5 2)(7 2) +
Int. J. Pure Appl. Sci. Technol., 15(2) (2013), 68-72 71 Case 3:. Γ(0.75 )Γ(1.5 ) = 2Γ(0.75) Γ(1.25 )Γ(1 ) (2.2) For <0.75: 1+ 9 16 (5 4) +2125 2048 (+1) (5 4)(9 4) +84375 32768 (+1)(+2) (5 4)(9 4)(13 4) + Case 4: = 2 2 Γ(0.75 )Γ(1.25 ) Γ(1 ) (2.3) For <0.5: 1+ 3 2 (5 4) +425 112 (+1) (5 4)(9 4) +28125 2464 (+1)(+2) (5 4)(9 4)(13 4) + = 4 2πΓ(0.5 )Γ(1.25 ) Γ(0.25) Γ(1 )Γ(0.75 ) (2.4) 3. Derivation and Analysis Without the loss of generality of the generalized Dixon s theorem (1.4), one can choose suitable numerator and denominator parameters in various ways to apply in (1.4) to get many convergent infinite series [7], [8], [9], and after a little algebraic manipulation can deduce all the listed results in section 2 and many more. For example if we put =,== and = in the left hand side as well as right hand side of (1.4) and applying the Pochhammer symbol or shifted factorials and the Gamma function expansions and the recurrence formulas listed in the introduction part are enough to get (2.1). Likewise by taking different combinations of the values of,, and and then applying in the similar fashion we can get all the remaining results listed in section 2. Let us discuss some of the special cases of each of the equation listed in section 2. Using =0.5 in (2.1) we get (1.1). Using =0.25 in (2.1) we get 1+ 1 50 + 1 216 + 5. + = 2704 4Γ(0.75) (3.1) Using =0.25,and 0.5 in (2.2) and after little algebraic manipulation we get (1.1) and (1.2). In (2.3) we can use =0.5 to get a Ramanujan type series for 1+ 3 32 + 17 5 27 5 32 7 2+ 32 77 2 + =4 (3.2)
Int. J. Pure Appl. Sci. Technol., 15(2) (2013), 68-72 72 And then using =0.25 we can get (1.3) 1+ 3 4 +17 5 4 + 81 5 2 2 4 + = (3.3) Γ(0.75) 1+9 1 4 +17 1 5 4 8 +25 1 5 9 4 8 12 + = (8+1) Γ(+0.25)!Γ(0.25) = 2 2 Γ(0.75) 2 2 Γ(0.75) At last but not the least we can apply =0.25 in (2.4) to get the same series (3.2) i.e., Ramanujan type series for. 5. Scope 1+ 3 32 + 17 5 27 5 32 7 2+ 32 77 2 + =4 Being open for further work by choosing different possible combinations one can generate many more important and curious results that will add to the existing literature of special functions and its applications. Acknowledgement The authors are highly grateful to the referee for carefully reading the manuscript and providing certain very useful suggestions which led to a better presentation of this research article. References [1] G. Arfken, Mathematical Methods for Physicists (3rd ed.), Orlando, FL: Academic Press, 1985. [2] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, Washing- ton, D.C., 1964, Reprinted by Dover Publication, New York, 1972. [3] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935. [4] B.C. Berndt, Ramanujan s Notebooks (Part-II), Springer Verlag, New York, 1987. [5] J.M. Borwein and P.B. Borwein, Class number three Ramanujan type series for, J. Comput. Appl. Math., 46(1993), 281-290. [6] E.D. Rainville, Special Functions, Macmillan, New York, 1960. [7] Salahuddin, Development of a summation theorem using Gauss theorem, Int. J. Pure Appl. Sci. Technol., 9(1) (2012), 52-60. [8] N. Shekhawat, On two summations due to Ramanujan and their generalizations, Advances in Computational Mathematics and its Applications, 2(1) (2012), 237-238. [9] N. Shekhawat, On a summation due to Ramanujan and its generalizations, Advances in Computational Mathematics and its Applications, 2(1) (2012), 243-244.