Analytical formulae for extended $_{3}F_{2}$-series of Watson–Whipple–Dixon with two extra integer parameters

Abstract

By combining the linearization method with Dougall's sum for well-poised 5 F 4-series, we investigate the generalized Watson series with two extra integer parameters. Four analytical formulae are established, which can also be used to evaluate the extended Whipple and Dixon series via the Thomae transformation. Twelve concrete formulae are presented as exemplification.

Loop equation analysis of the circular β ensembles

“…In order to compute the moments m k we expand the hypergeometric sum, integrate term-by-term and find an integral of the form (5.13), but with the replacements M → M + l/2 and k → k − M + l/2 for some l ∈ Z ≥0 . Resumming this again we have (5.19 (24) of [12] we see that what we seek is W 2k,0 (a, b, c) with a = −2M, b = 1 + β, c = −β (of course β = 2M but we only apply the termination through one parameter initially). Thus we can utilise Theorem 5, pg.…”

Variations on a hypergeometric theme

“…Note that the 4 F 3 (1) on the right-hand side of (4.2) is almost related to h(a) by the operation a → a + 1 -see (8.3). This result is particularly interesting because the 3 F 2 (1) that appears in (4.2) is a limiting case contiguous to Watson's theorem [16] and therefore can be evaluated in closed form:…”

“…In the case that d = −c − n, the 3 F 2 (1) on the right-hand side of (7.4) is easily identified (Note: here only, n ≥ −2 ) as being contiguous to Watson's theorem ([17, entry 1]) for which general closed forms are well known [16]. However, because the top and bottom parameters are almost all separated by integers, this becomes a very complicated general limiting case; when the limits are all evaluated, the result contains far too many terms to realistically reproduce here.…”

Summation formulae forq-Watson type4ϕ3-series

“…Inspired by the method due to Chu [3], we shall establish the following two families of summation formulae for q-Watson type 4φ3-series:…”