The q-analogue of stirling’s formula
F.H. Jackson defined a q-analogue of the factorial n! = 1∙2∙3 ⋯ n as (n!)q = 1∙ (1 + q) ∙ (1 + q + q2) ⋯ (1 + q + q2 + ⋯ +qn-1) which becomes the ordinary factorial as q → 1. He also defined the q-gamrna function as (FORMULA PRESENTED) and (FORMULA PRESENTED) where (FORMULA PRESENTED) It is known that if q → 1, Γq(x)) → Γ(x), where Γ(x) is the ordinary gamma function. Clearly Γq(n + 1) = (n!)q, so that the q-gamma function does extend the q factorial to non integer values of n. We will obtain an asymptotic expansion of Γq(z) as |z| →∞ in the right halfplane, which is uniform as q →1, and when q → 1, the asymptotic expansion becomes Stirling’s formula. © 1984 Rocky Mountain Mathematics Consortium.
Rocky Mountain Journal of Mathematics
The q-analogue of stirling’s formula.
Rocky Mountain Journal of Mathematics,
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