Gauss's Summation Theorem Research Articles

Further Ramanujan-like series containing harmonic numbers and squared binomial coefficients

The Gauss summation theorem and an extended $$_3F_2$$ -series of Watson and Whipple type are examined by means of power series expansions. Numerous Ramanujan-like series involving harmonic numbers and squared central binomial coefficients are evaluated in closed forms. Several remarkable identities discovered by Campbell (Ramanujan J 46(2):373–387, 2018) are included as particular cases.

On Some Series Identities for $\frac{1}{\Pi}$

By employing classical Watson's and Whipple's $_3F_{2}$-summation theorems, recently Liu, et al. have obtained a few Ramanujan type series for $\frac{1}{\pi}$ and deduced twelve interesting formulas for $\frac{1}{\pi}$. The aim of this short research paper is to point out that these twelve interesting formulas for $\frac{1}{\pi}$ can be easily obtained by employing classical Gauss's summation theorem.

Another Method for Deriving two Results Contiguous to Kummer's Second Theorem

The aim of this research paper is to derive two results closely related to the well known classical and useful Kummer's second theorem obtained earlier by Kim et al. [Comput. Math. & Math. Phys., 50 (3) (2010), 387 - 402] by employing classical Gauss's summation theorem for the series $_{2}F_{1}$.

On a new class of summation formulae involving the Laguerre polynomial

By elementary manipulation of series, a general transformation involving the generalized hypergeometric function is established. Kummer’s first theorem, the classical Gauss summation theorem and the generalized Kummer summation theorem due to Lavoie et al. [Generalizations of Whipple’s theorem on the sum of a 3 F 2, J. Comput. Appl. Math. 72 (1996), pp. 293–300] are then applied to obtain a new class of summation formulae involving the Laguerre polynomial, which have not previously appeared in the literature. Several related results due to Exton have also been given in a corrected form.

Contiguous Extensions of Dixon's Theorem on the Sum of a 3F2

In 1994, Lavoie et al. have succeeded in artificially constructing a formula consisting of twenty three interesting results, except for five cases, closely related to the classical Dixon's theorem on the sum of a by making a systematic use of some known relations among contiguous functions. We aim at presenting summation formulas for those five exceptional cases that can be derived by using the same technique developed by Bailey with the help of Gauss's summation theorem and generalized Kummer's theorem.

C n and D n Very-Well-Poised 10 φ 9 Transformations

In this paper we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised 10 \(\phi\) 9 transformation. We work in the setting of multiple basic hypergeometric series very-well-poised on the root systems A n , C n , and D n . Following the distillation of Bailey's ideas by Gasper and Rahman [11], we use a suitable interchange of multisums. We obtain C n and D n 10 \(\phi\) 9 transformations combined with A n , C n , and D n extensions of Jackson's 8 \(\phi\) 7 summation. Milne and Newcomb have previously obtained an analogous formula for A n series. Special cases of our 10 \(\phi\) 9 transformations include several new multivariable generalizations of Watson's transformation of an 8 \(\phi\) 7 into a multiple of a 4 \(\phi\) 3 series. We also deduce multidimensional extensions of Sears' 4 \(\phi\) 3 transformation formula, the second iterate of Heine's transformation, the q -Gauss summation theorem, and of the q -binomial theorem.

New hypergeometric transformations

The elementary manipulation of series is applied to obtain a quite general transformation involving hypergeometric functions. Hypergeometric identities not previously recorded in the literature are then deduced by means of Gauss's summation theorem and other hypergeometric summation theorems.

A q-analog of the Gauss summation theorem for hypergeometric series in U( n)

In this paper we derive a q-analog of Holman's U( n) generalization of the Gauss summation theorem. This result is obtained by iterating our mutivariable generalization of the Gauss reduction formula for basic hypergeometric series in U( n). Multiple series generalizations of both classical q-analogs of the Vandermonde summation theorem appear as special cases. We obtain the corresponding theorems for ordinary series by taking the limit as q → 1 of all these results.