Abstract
A procedure for generating infinite series identities makes use of the generalized method of exhaustion by analytically evaluating the inner series of the resulting double summation. Identities are generated involving both elementary and special functions. Infinite sums of special functions include those of the gamma and polygamma functions, the Hurwitz Zeta function, the polygamma function, the Gauss hypergeometric function, and the Lerch transcendent. The procedure can be automated with Mathematica (or equivalent software).
Highlights
The generalized method of exhaustion [5] provides series expansions for Riemann integrals of the form bN 2n−1 f (x)dx = (b − a) lim (−1)m+12−nf a N→∞ n=1 m=1 a + m(b − 2n a) ∞ 2n−1 = (b − a)(−1)m+12−nf n=1 m=1 a (1.1)The first expression is identical to the limit of the Riemann sum of order 2N for f (x) over [a, b], except for the missing endpoints f (a) or f (b), which can be neglected when N → ∞
Analytical sums are occasionally expressible in terms of elementary functions or in terms of special functions
As N → ∞, (3.2) is valid for all Riemann integrable functions. Each of these expressions leads to distinct identities, once the inner finite series is summed
Summary
A procedure for generating infinite series identities makes use of the generalized method of exhaustion by analytically evaluating the inner series of the resulting double summation. Identities are generated involving both elementary and special functions. Infinite sums of special functions include those of the gamma and polygamma functions, the Hurwitz Zeta function, the polygamma function, the Gauss hypergeometric function, and the Lerch transcendent. The procedure can be automated with Mathematica (or equivalent software).
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