A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas

A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curious series identities.


Introduction
Some of the most recent developments are on the use of different techniques for obtaining sums of hypergeometric series.In this paper, we present a new method for calculating the following summations and also a generalized theorem related to these series is investigated.We investigated the following summations formula with some restrictions for the functions (, ): We assume that the function ( − , ) has no poles at ( −  + ), where  is an integer ranged from 1 ≤  ≤ .
As it turns out, the above summation formula for the constant function (, ) = 1 gives us a new generalized hypergeometric transformation formula of the type ]                 = . ( We further investigated some  series closely related to the famous Roger-Ramanujan identities.
International Journal of Combinatorics In our present investigation dealing with the series identity, we shall also make use of other such higher transcendental functions, the Riemann Zeta function and Hurwitz Zeta function, which are defined by where  ()  is the generalized harmonic numbers defined by −1) (1) −  (−1) ( + 1)] ; for  ≥ 1,  ≥ 0. ( In the above identity we used the  notation for denoting the generalized polygamma function of order  which is given by the  + 1th times logarithmic derivative of Gamma function  () () =      (1) () =  +1  +1 ln Γ () ;  (1) = −. ( denotes the famous Euler-Mascheroni constant.The derivatives of generalized harmonic number are given: ()  = (−1) +1 Γ ( + ) Γ ()  ( + ,  + 1) .(7) And we used () V for the Pochhammer symbol defined (for , V ∈  and in terms of the Gamma function) by () V = Γ ( + V) Γ () =  ( + 1) ⋅ ⋅ ⋅ ( + V − 1) ; The generalized hypergeometric function is defined by The notation for generalized hypergeometric functions was introduced by Pochhammer in 1870 and modified by Barnes [1] and later by Maier and Slater [2].A number of notational variations are commonly used.Most common notation is introduced by Graham et al. [3] using square brackets and a semicolon.
The complete Bell polynomials of order  are defined as First few terms of these polynomials can be derived by

Main Results
Theorem 1.For integers  > 0,  ⩾ 0, where ( − , ) and Proof.Let us define a function   (, , ) where , ,  ∈ R such that We also assume the function ( − , ) has no poles at ( −  + ) for each  and  such that We introduce another notation for the th derivative with respect to  of the defined function   (, , ): Furthermore, we see the series also defined for .It can be extended to the interval [0, ∞], because when  → ( + ), where  is a positive integer, the further terms of the series just vanish. lim Making use of certain special properties of Bell polynomials we can evaluate successive derivative of a given function.Let us consider a function (, , ) which has a Taylor series expansion around ; the detailed procedure of these kinds is extensively discussed in the paper [4].Following the same process discussed in [5,6] we can write the following elegant identity by virtue of Bell polynomials: International Journal of Combinatorics 3 The following identity can be recovered from the same method used in [5, page 8]: represents the th order derivative harmonic number  (1)    with respect to .The derivative of harmonic numbers can be evaluated by using the formula given in Hence, upon considering ( 18) and ( 19), we find The above identity was derived extensively in [5] and a modified version was calculated in [7].Now differentiating (13)  times with respect to  and considering the case  = , where  is a positive integer as well as using the definition from (15), we can derive Making use of the identity derived in (20) we find From the asymptotic properties of Bell polynomials we have Considering the above property in (23), for  ≥ 1 and ,  ∈  + are both positive integers, we can readily evaluate the following limits: lim (1)   −  (1)  − ) , − Γ (2) ( (2)   −  (2)  − )] = 0. ( Considering next few cases, we can move to final relation lim where   is defined as in (19) Furthermore, expanding the product of generalized harmonic numbers in asymptotic form with Big  notation, we can further deduce Substituting the above asymptotic identity in (26) we find lim International Journal of Combinatorics After some tedious calculations we find lim In view of the known identity for Bell polynomials, We can finally obtain a closed form for the limit lim Substituting the limiting value of ( 26) in (31), we can readily obtain a closed form for the required limit lim Applying the above limiting value, we have Finally by combining ( 22) and (33), we finally obtain Theorem 1.
Theorem 2. For integers  > 0,  ⩾ 0, where Proof.The proof is similar to the previous one.Similarly as before we consider the function Then by the same process and with same restrictions we can easily obtain Theorem 2.
Differentiating (45) two times with respect to  and using the formula involving differentiation of generalized harmonic numbers stated in the first section, Finally substituting the expressions of ( 45) and ( 46) in (44), we easily obtain Theorem 5.
Corollary 16.For integers  ≥ 0, International Journal of Combinatorics 9 Proof.Substituting  = 2 and  = −1 in Theorem 3 and using the property of Bell polynomials, (2)  2 (, −1) Considering another classical result illustrated in Gould's book [13] Volume 5 which first appeared in American Math Monthly, for the case  = , we have Hence we have Substituting the value of  (2)  2 (, −1) from ( 74) in (72) we can immediately obtain Corollary 16.Various special cases of Corollary 16 can be found in several works of Wimp [20,21].
Some  Series Identities.-Pochhammer symbol, also called shifted factorial, is a -analog of the common Pochhammer symbol.It is defined as And the  binomial coefficients also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients are Let us define The operator , used extensively in several references, was recently used by authors in [3,8,22].
Now by the properties of Bell Polynomials we have We can further deduce This implies From the geometric series we have Applying the operator  on both sides  times, By means of this identity we have . (100) Using the previous identity (55) for Laguerre Polynomials and considering Theorem 2 for  = −,  = 1; () = 1/!we can easily derive Corollary 10.