Related papers: One-parameter Generalizations of Ramanujan's Formu…
The Hardy-Ramanujan formula for the number of integer partitions of $n$ is one of the most popular results in partition theory. While the unabridged final formula has been celebrated as reflecting the genius of its authors, it has become…
Let $L(s,\pi)=\sum_{n=1}^{\infty}\lambda_{\pi}(n)n^{-s}$ be an $L$-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $\lambda_{\pi}(n)$ strongly repel the low-lying zeros of…
We observe that certain famous evaluations of the Rogers-Ramanujan continued fraction $R(q)$ are close to $2\pi-6$ and $(2\pi-6)/2\pi$, and that $2\pi-6$ can be expressed by a Rogers-Ramanujan continued fraction in which $q$ is very nearly…
Via symbolic computation we deduce 97 new type series for powers of $\pi$ related to Ramanujan-type series. Here are three typical examples: $$\sum_{k=0}^\infty \frac{P(k) \binom{2k}k\binom{3k}k…
General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new…
We derive the mean square of the divisor function using only elementary techniques.
We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities,…
In this article we present ways to evaluate certain sums, products and continued fractions using tools from the theory of elliptic functions. The specific results appear to be new, although similar ones can be found in the leterature; in…
In a well-known 1914 paper, Ramanujan gave a number of rapidly converging series for $1/\pi$ which are derived using modular functions of higher level. D. V. and G. V. Chudnovsky in their 1988 paper derived an analogous series representing…
In this article we prove some identities which allow us to evaluate some multiple unit square integrals. In our examples we will give the value of some double and triple integrals. Then, we prove several classical integral formulas with the…
Carlitz proved a few generalizations of Mehler's formula. Later, Srivastava et al. gave a new proof for some extensions of Carlitz's formula. Here, a direct proof of the further generalization is given.
We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.
We generalize Warnaar's elliptic extension of a Macdonald multiparameter summation formula to Riemann surfaces of arbitrary genus.
In this paper we derive some asymptotic formulas for the $q$-Gamma function $\Gamma_{q}(z)$ for $q$ tending to 1.
We consider the limit periodic continued fractions of Stieltjes $$ \frac{1}{1-} \frac{g_1 z}{1-} \frac{g_2(1-g_1) z}{1-} \frac{g_3(1-g_2)z}{1-...,}, z\in \mathbb C, g_i\in(0,1), \lim\limits_{i\to \infty} g_i=1/2, \quad (1) $$ appearing as…
We prove a polynomial continued fraction identity for the constant $-\pi/4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a…
A simple integration by parts and telescopic cancellation leads to a rigorous derivation of the first 2 terms for the error in Ramanujan's asymptotic series for the nth partial sum of the harmonic series. Then Kummer's transformation gives…
We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish the higher derivatives estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow.…
We state and prove a claim of Ramanujan. As a consequence, a large class of Saalchutzian hypergeometric series is summed in closed form.
We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plucker formulas, addressing the total inflection of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety.…