A simple proof of Ramanujan’s ₁𝜓₁ sum

A simple proof of Ramanujan’s $_{1}\psi _{1}$ sum
HTML articles powered by AMS MathViewer

by Mourad E. H. Ismail PDF

Proc. Amer. Math. Soc. 63 (1977), 185-186 Request permission

We give a simple proof of the $_1{\psi _1}$ sum using basic hypergeometric functions.

References

George E. Andrews, On Ramanujan’s summation of $_{1}\psi _{1}(a;\,b;\,z)$, Proc. Amer. Math. Soc. 22 (1969), 552–553. MR 241703, DOI 10.1090/S0002-9939-1969-0241703-4
George E. Andrews, On a transformation of bilateral series with applications, Proc. Amer. Math. Soc. 25 (1970), 554–558. MR 257413, DOI 10.1090/S0002-9939-1970-0257413-1
George E. Andrews, Number theory, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1971. MR 0309838

A simple proof of Ramanujan’s summation of the

Wolfgang Hahn, Beiträge zur Theorie der Heineschen Reihen. Die $24$ Integrale der Hypergeometrischen $q$-Differenzengleichung. Das $q$-Analogon der Laplace-Transformation, Math. Nachr. 2 (1949), 340–379 (German). MR 35344, DOI 10.1002/mana.19490020604
M. Jackson, On Lerch’s transcendant and the basic bilateral hypergeometric series ${}_2\Psi _2$, J. London Math. Soc. 25 (1950), 189–196. MR 36882, DOI 10.1112/jlms/s1-25.3.189
Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688