Summing the hypergeometric function F 1 2 in the left member of (4.22) with the help of the Gauss summation formula (4.7) gives the result.
In this research, the authors discussed the asymptotic expansion of K 2 / ( λ m 2 + K 2 ) for large and small K, respectively, with the help of series summation formulas.
It is now easy to see that the F 2 3 on the right-hand side of (3.10) can be evaluated with the help of the extension of Kummer's summation theorem (1.9); we get e − x ∑ n = 0 ∞ ( ν + 1 ) n ( 1 − ν ) n x n n !
Our results presented here are derived with the help of two general summation formulae for the terminating F 1 2 ( 2 ) series which were very recently obtained by Kim et al. [25] and also include the relationship between F A ( 3 ) ( x, y, z ) and X 8 due to Exton [17].
Our results presented here are derived with the help of two general summation formulae for the terminating F 1 2 ( 2 ) series which were very recently obtained by Kim et al. and also include the relationship between F A ( 3 ) ( x, y, z ) and X 8 due to Exton.
Now, it is readily seen that the F 2 3 on the right-hand side of (3.6) can be evaluated with the help of the contiguous extension of Kummer's summation theorem in (1.10) to yield e − x ∑ n = 0 ∞ x n n !
The following inequalities could be deduced by using Young's inequality and norm inequalities with the help of changing the order of summations or exchanging the indices of the summations: (58).
The F 1 2 on the right-hand side can now be evaluated with the help of generalized Bailey summation theorem (1.7), and after some simplification, we easily arrive at the right-hand side of (2.2).
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.
Now we observe that the first F 1 2 on the right-hand side of (1.11) can be evaluated with the help of the known result (1.6) while the second F 1 2 on the right-hand side of (1.11) can be evaluated with the help of Kummer's summation theorem (1.5), after some simplification, we easily arrive at the right-hand side of (1.10).
