An Analogue of Ramanujan's Master Theorem

John Campbell , Sanjar Abrarov (1)

1 York University [Toronto]

We introduce an analogue of Ramanujan's Master Theorem that greatly generalizes some recent results on the construction of $\frac{1}{\pi}$ series. Implementing this technique through the use of computer algebra systems produces ``proof signatures'' for closed-form evaluations for new classes of infinite series and definite integrals. Using this integration method, we offer symbolic computations for a variety of new series, including new binomial-harmonic series for $\frac{1}{\pi}$, such as the elegant series $$\sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n H_{2 n}}{16^n (n+1)} = 4-\frac{7 \pi }{3}+8 \ln (2)+\frac{48 \ln ^2(2) - 16 G-16 \ln (2)}{\pi }$$ introduced in our article, letting $G$ denote Catalan's constant, as well as new closed-form evaluations for ${}_{3}F_{2}(1)$ series, as in the equality $$ {}_{3}F_{2}\left[ \begin{matrix} \frac{1}{4},\frac{5}{4},\frac{3}{2} \\ 1, \frac{9}{4} \end{matrix} \ \Bigg| \ 1 \right] = \frac{\Gamma^2 \left(\frac{1}{4}\right) \left(15 \sqrt{2}-5 \ln \left(1 + \sqrt{2}\right)\right)}{16 \pi ^{3/2}} $$ introduced in our article.

Domaines: Mathématiques [math]; Analyse classique [math.CA]

John Campbell, Sanjar Abrarov. An Analogue of Ramanujan's Master Theorem. 2018. ⟨hal-01897255⟩