Explicit and asymptotic formulae for Vasyunin-cotangent sums (original) (raw)

A q-continued fraction

2006

Let a, b, c, d be complex numbers with d 6= 0 and |q| \u3c 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq2n+1 + cqn (a + b)q n+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d) j (−c/bd)j q j(j+3)/2 (q)j (−aq2/d)j P∞ j=0 (b/d) j (−c/bd)j q j(j+1)/2 (q)j (−aq/d)j . We then use this result to deduce various corollaries, including the following: 1 1 − q 1 + q − q 3 1 + q 2 − q 5 1 + q 3 − · · · − q 2n−1 1 + q n − · · · = (q 2 ; q 3 )∞ (q; q 3)∞ , (−aq)∞ X∞ j=0 (bq) j (−c/b)j q j(j−1)/2 (q)j (−aq)j = (−bq)∞ X∞ j=0 (aq) j (−c/a)j q j(j−1)/2 (q)j (−bq)j , and the Rogers-Ramanujan identities, X∞ n=0 q n 2 (q; q)n = 1 (q; q 5)∞(q 4; q 5)∞ , X∞ n=0 q n 2+n (q; q)n = 1 (q 2; q 5)∞(q 3; q 5)∞

Some identities for Ramanujan-Gollnitz-Gordon continued fraction

Australian Journal of Mathematical Analysis and Applications

In this paper, we obtain certain PPP--$Q$ eta--function identities, using which we establish identities providing modular relations between Ramanujan-G\"{o}llnitz-Gordon continued fraction H(q)H(q)H(q) and H(qn)H(q^n)H(qn) for n=2,3,4,5,7,8,9,11,13,n= 2, 3, 4, 5, 7, 8, 9, 11, 13, n=2,3,4,5,7,8,9,11,13, 15,15,15, $ 17,$ $ 19, $ 232323, 252525, 292929 and 555555.

Representations of Ramanujan Continued Fraction in Terms of Combinatorial Partition Identities

Honam Mathematical Journal, 2016

Adiga and Anitha [1] investigated the Ramanujan's continued fraction (18) to present many interesting identities. Motivated by this work, by using known formulas, we also investigate the Ramanujan's continued fraction (18) to give certain relationships between the Ramanujan's continued fraction and the combinatorial partition identities given by Andrews et al. [3].

A survey of results on a generalization of Ramanujan sum

Ramanujan sum is a sum of powers of primitive roots of unity defined by Srinivasa Ramanujan. After this, there have been many generalizations of the Ramanujan sum one of which was given by E. Cohen. In a series of articles, he proved that several interesting properties of the classical Ramanujan sum extends to his generalization as well. Many other authors followed the footsteps of Cohen to give various such generalized results. In this survey article, we list some of the most important properties of the original sum and the generalization and also give some expected results using the generalized sum