New series for powers of <inline-formula><tex-math id="M1">$ \pi $</tex-math></inline-formula> and related congruences

Abstract

Via symbolic computation we deduce 97 new type series for powers of π related to Ramanujan-type series. Here are three typical examples: ∞ k=0 ∞ k=0 39480k + 7321 (−29700) k T k (14, 1)T k (11, −11) 2 = 6795 √ 5 π. Eighteen of the new series in this paper involve some imaginary quadratic fields with class number 8.

Some new series for $1/\pi $ motivated by congruences

“…In a recently published paper [17] the author proposed four conjectural series for 1/π of a new type: Conjecture 4.1. We have the following identities:…”

“…(ii) Let p be an odd prime with p ̸ = 7. Then Moreover, when p ≡ 1 (mod 12), for any n ∈ Z + the number (17,16) divided by (pn) 2 2n n is a p-adic integer. (iii) For any prime p > 7, we have divided by (pn) 2 2n n is a p-adic integer.…”

“…(with p any prime greater than 3) which has nothing to do with the Legendre symbol −3 p . For the author's philosophy to generate series for 1/π via congruences, one may consult the survey [12] and the recent paper [17,Section 1].…”

Using Beta Function and Factorization to Find New π-Series

“…In [10], Sun derived several identities involving π by the telescoping method. For example, from Bauer's series [2] ∞ ∑ k=0 (4k + 1)…”