S. Bhargava

University of Mysore | mysore · Department of Mathematics

If Dn(π1,n,π2,n,⋯,π n,n):= (Equation Presented) is the multinomial representation of the product π (ai + aj)/1≤i<j≤n in terms of elementary symmetric functions π1,n, π2,⋯, πn,n of the variables a1, a2,⋯, an, we obtain a recurrence relation for Dn(π 1,n,π2,n,π3,n,⋯,π n,n). With the help of this recurrence relation, we evolve four immediate successor...

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q-binomial theorem.

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1\psi 1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q-binomial theorem.

We obtain a modular transformation for the theta function å - ¥¥ å - ¥¥ qa( m2 + mn ) + cn2 + lm + mn + nV Am + BnZCm + Dn, \sum\limits_{ - \infty }^\infty {\sum\limits_{ - \infty }^\infty {{q^{a\left( {{m^2} + mn} \right) + c{n^2} + \lambda m + \mu n + {\nu_\varsigma }Am + B{n_Z}Cm + Dn}}}, } which enables us to unify and extend several modular...

We obtain a convolution identity for the coefficients B n (α,θ,q) defined by ∑ n=-∞ ∞ B n (α,θ,q)x n =∏ n=1 ∞ (1+2xq n cosθ+x 2 q 2n ) ∏ n=1 ∞ (1+αq n xe iθ ), using the well-known Ramanujan’s 1 ψ 1 -summation formula. The work presented here complements the works of K.-W. Yang, S. Bhargava, C. Adiga and D. D. Somashekara and of H. M. Srivastava.

We establish in a simple self contained manner a triple product identity for the cubic theta function a(q,ζz) = ∑m = -∞∞ ∑n = -∞∞ q m2+mn+n2 ζn+m zn-m. We also determine on our way a two-parameter family of zeros of a (q, ζz).

We establish some interesting q-gamma and q-beta function identities as an application of ‘Ramanujan’s’ Reciprocity theorem.

In this paper we obtain a class of modular equations in Ramanujan’s alternative theory of elliptic functions of signature 4 and employ them to obtain a new class of P-Q eta-function identities with four moduli akin to Ramanujan’s.

In this work we obtain some new algebraic relations between the functions of the type. $P_{m,n} (z):=η(nz) η(z) / η(mnz) η(mz)$ or $η(nz) η(mz) / η(mnz) η(z) $ and $Q_{m,n} (z):=P_{m,n} (2z)$, where m and n are certain positive integers and η(z) stands, as usual, for the Dedekind’s eta-function. The proofs are elementary in the spirit of Ramanujan...

I t i s w ell known thaRamanujan's remarkable summation formula' uniies and gen-eralizes the q-binomial theorem and the triple product identity and has numerous applications. In this note we will demonstrate how, after a suitable transformation of the series side, it can belooked upon as a 2-parameter generalization of the quintuple product identit...

As a limiting case of Ramanujan’s 1 Ψ 1 summation we obtain a generalization of Jacobi’s expansion for (q) ∞ 3 .

this paper is written for lowbrows. Only elementary algebra is needed to prove the lion's share of theorems reported here. Most are found in the unorganized portion of Ramanujan's second notebook, his third notebook, and problems that he posed for readers of the Journal of the Indian Mathematical Society. The results we describe fall under the head...

In this note we obtain a formula for the number of representations of an integer n(≥1) in the form k 2 +5m 2 . Our results follow from Andrews’ generalization of the well-known Ramanujan’s 1 Ψ 1 summation [Adv. Math. 41, 137–172 (1981; Zbl 0477.33001)].

In his famous paper on modular equations and approximations to π, Ramanujan offers several series representations for 1/π, which he claims are derived from “corresponding theories” in which the classical base q is replaced by one of three other bases. The formulas for 1/π were only recently proved by J. M. and P. B. Borwein in 1987, but these “corr...

We obtain an interesting 2ψ2 summation formula and demonstrate its diverse uses leading to some (i) sums of squares theorems (ii) Ramanujan's Fourier series developments related to theta-functions (iii) Lambert series identities related to Dedekind eta-function (iv) q-gamma and g-beta identities.

In this note we deduce a family of eta-function identities using Ramanujan′s 1ψ1 summation.

In the unorganized portions of his second notebook, Ramanujan states without proofs 10 inversion formulas for the lemniscate integral and two similar integrals. These 10 formulas are proven here.

If a and n are positive integers and if is the greatest integer function we obtain upper and lower estimates for stated by Ramanujan in his notebooks.

In this note we establish continued fraction developments for the ratios of the basic hypergeometric function2ϕ1(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several continued fraction identities including those of Srinivasa Ramanujan.

We obtain here a variational principle characterizing an optimal filter. This variational principle is dual to the one obtained by Berkovitz and Pollard in Ref. 2.

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