Abstract
We establish two recurrence relations for some Clausen’s hypergeometric functions with unit argument. We solve them to give the explicit formulas. Additionally, we use the moments of Ramanujan’s generalized elliptic integrals to obtain these recurrence relations.
Highlights
The hypergeometric function [1] is defined to be the complex analytic function∑ p+1 Fp a1, a2, . . ., ap+1 b1, b2, . . ., bp z = ∞ n=0(a1)n(a2)n · (b1 )n (b2 )n · ·(ap+1)n ·n zn n!, where (α)nΓ(α + n) Γ(α) α(α + 1) ·
It is interesting to give an explicit formula for the corresponding general form
We provide two recurrence relations for the hypergeometric series F(x)
Summary
The (generalized) hypergeometric function [1] is defined to be the complex analytic function. Denotes the The Pochhammer symbol, Γ(s) is the Euler’s Γ-function, p is a non-negative integer, the complex numbers ai, bj are called, respectively, the numerator and denominator parameters, and z is called the variable. {1, 2, 3}, and k ∈ {1, 2, 3, 4} by applying their method to the elliptic fibration y2 = 2x3 − 3x2 + t where = 2, 3, 4, 5, respectively Motivated by their works, it is interesting to give an explicit formula for the corresponding general form. For any non-negative integer n, we have the following explicit formulas:. K2x−1, In the last section, we will use the moments of Ramanujan’s generalized elliptic integral to give another method of obtaining the explicit evaluations. We use the moments of Ramanujan’s generalized elliptic integral to give another method of obtaining the same evaluations
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