Generalizations Of Ramanujan Research Articles

Koshliakov zeta functions I: Modular relations

We examine an unstudied manuscript of N.S. Koshliakov over 150 pages long and containing the theory of two interesting generalizations ζp(s) and ηp(s) of the Riemann zeta function ζ(s), which we call Koshliakov zeta functions. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each p∈R+. The first one is a generalization of Ramanujan's famous formula for ζ(2m+1) and the second is an elegant extension of a modular relation on page 220 of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for ζ(4m+3).

Generalized Lambert series and arithmetic nature of odd zeta values

AbstractIt is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.

GENERALIZED LAMBERT SERIES, RAABE’S COSINE TRANSFORM AND A GENERALIZATION OF RAMANUJAN’S FORMULA FOR

A comprehensive study of the generalized Lambert series$\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$,$N\in \mathbb{N}$and$h\in \mathbb{Z}$, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for$\unicode[STIX]{x1D701}(2m+1)$for$m>0$, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for$\unicode[STIX]{x1D701}(1/N),N$even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series$E_{2}(z)$and$E_{2}(-1/z)$. An identity relating$\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$is obtained for$N$odd and$m\in \mathbb{N}$. In particular, this gives a beautiful relation between$\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$and$\unicode[STIX]{x1D701}(11)$. New results involving infinite series of hyperbolic functions with$n^{2}$in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.

A generalization of Ramanujan's congruence to modular forms of prime level

We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin–Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup Γ0(N), and on a version of Ihara's lemma.

On Schwarzian Triangle Functions, Automorphic Forms and a Generalization of Ramanujan’s Triple Differential Equations

Let G be the group of the fractional linear transformations generated by $$T(\tau ) = \tau + \lambda ,S(\tau ) = \frac{{\tau \cos \frac{\pi }{n} + \sin \frac{\pi }{n}}}{{ - \tau \sin \frac{\pi }{n} + \cos \frac{\pi }{n}}};$$ where $$\lambda = 2\frac{{\cos \frac{\pi }{m} + \cos \frac{\pi }{n}}}{{\sin \frac{\pi }{n}}};$$ m, n is a pair of integers with either n ≥ 2,m ≥ 3 or n ≥ 3,m ≥ 2; τ lies in the upper half plane H.

Concerning an infinite series of Ramanujan related to the natural logarithm

We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.

Ramanujan's cubic transformation inequalities for zero-balanced hypergeometric functions

In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.

GENERALIZATIONS OF RAMANUJAN’S RANK FUNCTIONS COLLECTED FROM RAMANUJAN’S LOST NOTEBOOK

In1916, Srinivasa Ramanujan defined the Mock Theta functions in his lost notebook and unpublished papers. We prove the Mock Theta Conjectures with the help of Dyson’s rank and S. Ramanujan’s Mock Theta functions. These functions were quoted in Ramanujan’s lost notebook and unpublished papers. In1916, Ramanujan stated the theta series in x like A(x), B(x), C(x), D(x). We discuss the Ramanujan’s functions with the help of Dyson’s rank symbols. These functions are useful to prove the Mock Theta Conjectures. Now first Mock Theta Conjecture is “The number of partitions of 5n with rank congruent to 1 modulo 5 equals the number of partitions of 5n with rank congruent to 0 modulo 5 plus the number of partitions of n with unique smallest part and all other parts the double of the smallest part”, and Second Mock Theta Conjecture is “The double of the number of partitions of with rank congruent to 2 modulo 5 equals the sum of the number of partitions of with rank congruent to 0 and congruent to1 modulo 5, and the sum of one and the number of partitions of n with unique smallest part and all other parts  one plus the double of the smallest part”. This paper shows how to prove the Theorem 1.3 with the help of Dyson’s rank symbols N(0,5,5n+1), N(2,5, 5n+1) and shows how to prove the Theorem 1.4 with the help of Ramanujan’s theta series and Dyson’s rank symbols N(1,5, 5n+2), N(2,5, 5n+2) respectively.

Ramanujan’s fifth order and tenth order mock theta functions - a generalization

A generalization of Ramanujan’s fifth order and tenth order mock theta functions is given. It is shown that these belong to the family of Fq-functions. Using the properties of Fq-functions, relationship is given between these generalized fifth order mock theta functions of the first group with the generalized functions of the second group. The same is done for the generalized functions of the tenth order. q-Integral representation and multibasic expansions are also given.

On the functional properties of the confluent hypergeometric zeta-function

A zeta-function associated with Kummer’s confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan’s formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.

Ramanujan type congruences for the Klingen-Eisenstein series

In the case of Siegel modular forms of degree \(n\), we prove that, for almost all prime ideals \(\mathfrak {p}\) in any ring of algebraic integers, mod \(\mathfrak {p}^m\) cusp forms are congruent to true cusp forms of the same weight. As an application we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan’s congruence. We will conclude by giving numerical examples.

PARTITION IDENTITIES FOR RAMANUJAN'S THIRD-ORDER MOCK THETA FUNCTIONS

We find two involutions on partitions that lead to partition identities for Ramanujan’s third order mock theta functions φ(−q) and ψ(−q). We also give an involution for Fine’s partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third order mock theta functions are consequences of these partition identities. Our combinatorial constructions also apply to Andrews’ generalizations of Ramanujan’s identities.

An approximation for zero-balanced Appell function [formula omitted] near [formula omitted

We suggest an approximation for the zero-balanced Appell hypergeometric function F 1 near the singular point ( 1 , 1 ) . Our approximation can be viewed as a generalization of Ramanujan's approximation for zero-balanced F 1 2 and is expressed in terms of F 2 3 . We find an error bound and prove some basic properties of the suggested approximation which reproduce the similar properties of the Appell function. Our approximation reduces to the approximation of Carlson–Gustafson when the Appell function reduces to the first incomplete elliptic integral.

Generalizations of Ramanujan's reciprocity theorem and their applications

First, we briefly survey Ramanujan's reciprocity theorem for a certain q-series related to partial theta functions and give a new proof of the theorem. Next, we derive generalizations of the reciprocity theorem that are also generalizations of the 1ψ1 summation formula and Jacobi triple product identity and show that these reciprocity theorems lead to generalizations of the quintuple product identity, as well. Last, we present some applications of the generalized reciprocity theorems and product identities, including new representations for generating functions for sums of six squares and those for overpartitions.

Integer Powers of Arcsin

New simple nested-sum representations for powers of the arcsin function are given. This generalization of Ramanujan's work makes connections to finite binomial sums and polylogarithms.

A New $A_n$ Extension of Ramanujan's ${}_1\psi_1$ Summation with Applications to Multilateral An Series

In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we provide several A_n extensions of Bailey's 2-psi-2 transformations, and several A_n extensions of a particular 2-psi-2 summation.

On the values of the Riemann zeta-function at rational arguments

In a companion paper, ``On multi Hurwitz-zeta function values at rational arguments, Acta Arith. {\bf 107} (2003), 45-67'', we obtained a closed form evaluation of Ramanujan's type of the values of the (multiple) Hurwitz zeta-function at rational arguments (with denominator even and numerator odd), which was in turn a vast generalization of D. Klusch's and M. Katsurada's generalization of Ramanujan's formula. In this paper we shall continue our pursuit, specializing to the Riemann zeta-function, and obtain a closed form evaluation thereof at all rational arguments, with no restriction to the form of the rationals, in the critical strip. This is a complete generalization of the results of the aforementioned two authors. We shall obtain as a byproduct some curious identities among the Riemann zeta-values.

On applications of hypergeometric functions

This is a survey article on the author's involvement over the years with hypergeometric functions. We discuss our counter-example to one of M. Robertson's conjectures, our results on the omitted values problems, Brannan's conjecture on the coefficients of a certain power series, generalizations of Ramanujan's asymptotic formulas for complete elliptic integrals and Muir's 1883 approximation to the arc length of an ellipse involving an inequality for some 3 F 2's.

Heine transformations for a new kind of basic hypergeometric series in U( n)

Heine transformations are proved for a new kind of multivariate basic hypergeometric series which had been previously introduced by Krattenthaler in connection with generating functions for nonintersecting lattice paths. As a consequence, a q-Gauss and q-Chu-Vandermonde sum are proved and also a generalization of Ramanujan's 1ψ 1 sum.

An identity for a class of arithmetical functions of two variables

For a positive integer r, let r∗ denote the quotient of r by its largest squarefree divisor (1∗ = 1). Recently, K. R. Johnson proved that urn:x-wiley:01611712:media:ijmm180307:ijmm180307-math-0001 according as r∗|n or not where C(n, r) is the well known Ramanujan′s sum. In this paper, using a different method, we generalize (∗) to a wide class of arithmetical functions of 2 variables and deduce as special cases (∗) and similar formulae for several generalizations of Ramanujan′′s sum.