Ramanujan Identities Research Articles

Legendre Theorems for a Class of Partitions with Initial Repetitions

AbstractPartitions with initial repetitions were introduced by George Andrews. We consider a subclass of these partitions and find Legendre theorems associated with their respective partition functions. The results in turn provide partition-theoretic interpretations of some Rogers–Ramanujan identities due to Lucy J. Slater.

Some separable integer partition classes

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus [Formula: see text]. We also extend separable integer partition classes with modulus [Formula: see text] to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.

On the q-binomial identities involving the Legendre symbol modulo 3

We use a polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 to prove six identities involving the q-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

A note on odd zeta values over any number field and extended Eisenstein series

In this article, we study transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. Our transformation formula for the Dedekind zeta function at odd integers leads to a new number field extension of Eisenstein series, which satisfies the transformation z↦−1/z like an integral weight modular form over SL2(Z). The results provide a number of important applications, which are important in studying the behavior of odd zeta values as well as Lambert series in an arbitrary number field.

Character analogues of Cohen-type identities and related Voronoï summation formulas

In [4], B. C. Berndt and A. Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, in a follow-up paper [20], S. Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified K-Bessel function in light of recent results obtained by the first author and B. Maji [1]. Moreover, we provide a new expression for L(1,χ) from which we establish the positivity of L(1,χ) for any real primitive character χ. In addition, we deduce Cohen-type identities and then exhibit the Voronoï-type summation formulas for them.

Ramanujan’s formula for odd zeta values: a proof by Mittag-Leffler expansion and applications

Using Mittag-Leffler expansion a novel proof of Ramanujan’s famous formula for zeta (2m-1) is presented. Here, the formula can be derived by Taylor series of a generating function and its Mittag-Leffler expansion. Furthermore, generalized formulas for the fast-converging series of Plouffe are offered. But Ramanujan’s equation not only produces series for odd zeta values, surprisingly it is shown that in the limit to its singularity it provides the identities for even zeta values, namely Euler’s classical formula. Finally, a new triangle identity of Ramanujan’s formula is presented, hinting that properties and symmetries of the equation are far from all uncovered. For this work, a new representation of the equations is introduced, that significantly simplifies analyses and applications.

On Ramanujan’s continued fractions of order twenty-four

Two continued fractions U(q) and V(q) of order twenty-four are obtained from a general continued fraction identity of Ramanujan. Some theta-function and modular identities for U(q) and V(q) are established to prove general theorems for the explicit evaluations of U(±q) and V(±q). From the theta-function identities of U(q) and V(q), three colour partition identities are derived as application to partition theory of integer. Further, 2-, 4- and 8-dissection formulas are established for the continued fractions U∗(q)≔q−5/2U(q) and V∗(q)≔q−1/2V(q), and their reciprocals.

Equivalent criterion for the grand Riemann hypothesis associated with Maass cusp forms

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated with the primitive Maass cusp form over $\Gamma _0(N)$.

Hurwitz zeta functions and Ramanujan's identity for odd zeta values

Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing that associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.

On some P-Q modular equations of degree 45

On page 330 of his second notebook, Srinivasa Ramanujan recorded a P-Q modular equation of degree 45, proof of which has been given by Bruce C. Berndt via theory of modular forms. We in this paper, give a simple proof of the same using the identities of Ramanujan and also establish few new P-Q modular equations of degree 45. Further using these, we establish certain new modular equations of signature 3.

Riesz-type criteria for L-functions in the Selberg class

AbstractWe formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$ .

On the identities of Ramanujan – a q-series approach

The idea of this paper is to prove two general theorems using q-series and deduce some partial theta identities and false theta identities. Some of the identities of Ramanujan come as special cases.

The mixed mock modularity of a new $U$-type function related to the Andrews-Gordon identities

In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new vector-valued $U$-type function. We obtain an expression for a corresponding family of Hecke--Appell-type sums in terms of mixed mock modular forms; that is, we express the sum in terms of Appell functions and theta functions. This $U$-type function appears from considering the special polynomials related to generating functions for the partitions occurring in Gordon’s generalization of the Rogers--Ramanujan identities.

An operational calculus generalization of Ramanujan's master theorem

We give a formal extension of Ramanujan's master theorem using operational methods. The resulting identity transforms the computation of an integral of a product on the half-line to the computation of a Laplace transform. Since the identity is purely formal, we show consistency of this operational approach with various standard calculus results, followed by several examples, including a Feynman diagram evaluation, to illustrate the power of the extension. We then briefly discuss the connection between Ramanujan's master theorem and identities of Hardy and Carr before extending the latter identities in the same way we extended Ramanujan's. Finally, we generalize our results, producing additional interesting identities as a corollary.

A NOTE ON ROGERS-RAMANUJAN-SLATER TYPE THETA FUNCTION IDENTITY

In this paper, we research theta function identity involving Rogers– Ramanujan identity and establish a Rogers–Ramanujan–Slater type theta function identity related to G(q) and φ(q).

Neighborly partitions and the numerators of Rogers–Ramanujan identities

We prove two partition identities which are dual to the Rogers–Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions), monomial ideals and some infinite graphs.

A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities

A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities

Eisenstein series of level 6 and level 10 with their applications to theta function identities of Ramanujan

S. Ramanujan recorded theta function identities of different levels in the unorganized pages of his second notebook and the lost notebook. In this paper, we prove level 6 and level 10 theta function identities by using Eisenstein series identities.

On two theta function identities of Ramanujan

On two theta function identities of Ramanujan

Dirichlet series under standard convolutions: variations on Ramanujan’s identity for odd zeta values

Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by Dixit et al., or the Fourier expansion of the convolution of Bernoulli–Barnes polynomials by Komori et al.