Ramanujan's Function Research Articles

Indefinite q-integrals from a method using q-Riccati equations

In an earlier work, a method was introduced for obtaining indefinite q-integrals of q-special functions from the second-order linear q-difference equations that define them. In this paper, we reformulate the method in terms of q-Riccati equations, which are nonlinear and first order. We derive q-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for the q-Airy function, the Ramanujan function, the discrete q-Hermite I and II polynomials, the q-hypergeometric functions, the q-Laguerre polynomials, the Stieltjes-Wigert polynomial, the little q-Legendre and the big q-Legendre polynomials.

On further modular relations for the Rogers–Ramanujan functions

On further modular relations for the Rogers–Ramanujan functions

Behavior of Binomial Distribution near Its Median

We study the behavior of the cumulative distribution function of a binomial random variable with parameters n and b{text{/}}(n + c) at the point b – 1 for positive integers b leqslant n and real c in [0,;1]. Our results can be applied directly to the well-known problem about small deviations of sums of independents random variables from their expectations. Moreover, we answer the question about the monotonicity of the Ramanujan function for the binomial distribution posed by Jogdeo and Samuels in 1968.

Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little q-Jacobi polynomial. From their large N form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.

On sum-of-tails identities

In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan and Liang, an identity of Ramanujan as well as two new results - one for Ramanujan's function σ(q) and another for the function recently introduced by Andrews and Ballantine. Later we introduce a new generalization FFWc(n) of a function of Fokkink, Fokkink and Wang and derive an identity for its generating function. This gives, as a special case, a recent representation for the generating function of spt(n) given by Andrews, Garvan and Liang. We also obtain some weighted partition identities along with new representations for two of Ramanujan's third order mock theta functions through combinatorial techniques.

Integral and series representations of q-polynomials and functions: Part III theta, Ramanujan and q-Bessel functions

In this paper, we use an identity connecting a modified [Formula: see text]-Bessel function and a [Formula: see text] function to give [Formula: see text]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [Formula: see text]-version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, [Formula: see text]-Lommel and [Formula: see text]-Bessel polynomials. We introduce and study [Formula: see text]-analogues of the spherical Bessel functions.

Modular relations for the Rogers–Ramanujan functions with applications to partitions

In this paper, we obtain several new modular relations for the Rogers–Ramanujan functions. Furthermore, we give partition theoretic interpretations for some of our modular relations.

On 5- and 10-dissections for some infinite products

Quite recently, Xia and Zhao established the 10-dissections for Hirschhorn’s two infinite q-series products by using two MAPLE packages and the theory of modular forms. Utilizing the Jacobi triple product identity, we not only establish the 10-dissections for two infinite q-series products, introduced by Baruah and Kaur, but give an elementary proof of the 10-dissections due to Xia and Zhao. Moreover, we obtain the 5-dissections for four quotients of infinite q-series products related to the Rogers–Ramanujan functions. Using these dissections, the coefficients in these series expansions have periodic sign patterns with a few exceptions.

On conjectures of Koike and Somos for modular identities for the Rogers–Ramanujan functions

In this paper, we prove modular identities conjectured by Koike and Somos for the Rogers–Ramanujan functions. Our methods focus on approaches that Ramanujan could have employed, including the theory of modular equations, dissections of identities, and methods derived from the approaches of Watson and Rogers.

On a Ramanujan type entire function and its zeros

In this paper, we derive some properties of a Ramanujan type entire function. A mild generalization of the Garret-Ismail-Stanton m-version of the Rogers-Ramanujan identities is obtained. Moreover, we investigate the zeros of the Ramanujan type entire function, and our results generalize those for the zeros of the Ramanujan function. Finally, an integral equation related to the Ramanujan type entire function is also derived.

Wronskians of theta functions and series for 1/π

In this article, we define functions analogous to Ramanujan's function f(n) defined in his famous paper “Modular equations and approximations to π”. We then use these new functions to study Ramanujan's series for 1/π associated with the classical, cubic and quartic bases.

𝑞-Bessel functions and Rogers-Ramanujan type identities

We evaluate q q -Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the m m -version of the Rogers-Ramanujan identities. We also prove several generating functions for Stieltjes-Wigert polynomials with argument depending on the degree. In addition we give several Rogers-Ramanujan type identities.

On a simple model of $$X_0(N)$$ X 0 ( N )

We find plane models for all $$X_0(N)$$ , $$N\ge 2$$ . We observe a map from the modular curve $$X_0(N)$$ to the projective plane constructed using modular forms of weight 12 for the group $$\Gamma _0(N)$$ ; the Ramanujan function $$\Delta $$ , $$\Delta (N\cdot )$$ and the third power of Eisestein series of weight 4, $$E_4^3$$ , and prove that this map is birational equivalence for every $$N\ge 2$$ . The equation of the model is the minimal polynomial of $$\Delta (N\cdot )/\Delta $$ over $${\mathbb {C}}(j)$$ .

Integral and series representations of q-polynomials and functions: Part I

By applying an integral representation for [Formula: see text], we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of [Formula: see text]-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include [Formula: see text]-Bessel functions, the Ramanujan function, Stieltjes–Wigert polynomials, [Formula: see text]-Hermite and [Formula: see text]-Hermite polynomials, and the [Formula: see text]-exponential functions [Formula: see text], [Formula: see text] and [Formula: see text]. Their representations are in turn used to derive many new identities involving [Formula: see text]-functions and polynomials. In this paper, we also present contour integral representations for the above mentioned functions and polynomials.

New modular relations involving cubes of the Göllnitz–Gordon functions

Chen and Huang established some elegant modular relations for the Gollnitz–Gordon functions analogous to Ramanujan’s list of forty identities for the Rogers–Ramanujan functions. In this paper, we derive some new modular relations involving cubes of the Gollnitz–Gordon functions. Furthermore, we also provide new proofs of some modular relations for the Gollnitz–Gordon functions due to Gugg.

New Modular Relations for the Ratio’s of Ramanujan Function X(−q) of degree 5

New Modular Relations for the Ratio’s of Ramanujan Function X(−q) of degree 5

Applications of the Heine and Bauer–Muir transformations to Rogers–Ramanujan type continued fractions

In this paper we show that various continued fractions for the quotient of general Ramanujan functions G(aq,b,λq)/G(a,b,λ) may be derived from each other via Bauer–Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer–Muir transformations converge to the same limit. We also show that these continued fractions may be derived from either Heine's continued fraction for a ratio of ϕ12 functions, or other similar continued fraction expansions of ratios of ϕ12 functions. Further, by employing essentially the same methods, a new continued fraction for G(aq,b,λq)/G(a,b,λ) is derived. Finally we derive a number of new versions of some beautiful continued fraction expansions of Ramanujan for certain combinations of infinite products, with the following being an example:(−a,b;q)∞−(a,−b;q)∞(−a,b;q)∞+(a,−b;q)∞=(a−b)1−ab−(1−a2)(1−b2)q1−abq2−(a−bq2)(b−aq2)q1−abq4−(1−a2q2)(1−b2q2)q31−abq6−(a−bq4)(b−aq4)q31−abq8−⋯.

On rational matrix exact covering systems of [formula omitted] and its applications to Ramanujan's forty identities

For a nonsingular integer matrix B, the set of cosets of the quotient module Zn/BZn forms an exact covering system (ECS) of Zn. In this paper, we use the Smith normal form to obtain another type of matrix ECS with rational entries which we call rational matrix ECS. Using rational matrix ECS of Z2, we prove eight identities in Ramanujan's list of forty identities for the Rogers–Ramanujan functions, as well as some other identities.

Some new modular relations for the Rogers–Ramanujan type functions of order eleven with applications to partitions

In this paper, we establish several modular relations for the Rogers–Ramanujan type functions of order eleven which are analogous to Ramanujan's forty identities for Rogers–Ramanujan functions. Furthermore, we give interesting partition-theoretic interpretation of some of the modular relations which are derived in this paper.

FACTORIALS AND THE RAMANUJAN FUNCTION

AbstractIn 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.