Computing an Order-Complete Basis for $$M^{\infty }(N)$$ and Applications

International Mathematics Research Notices

Compositio Mathematica, 2004

We investigate the arithmetic and combinatorial significance of the values of the polynomials j n (x) defined by the q-expansion ∞ n=0 j n (x)q n := E 4 (z) 2 E 6 (z) ∆(z) • 1 j(z) − x. They allow us to provide an explicit description of the action of the Ramanujan Thetaoperator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas.

2016

Many generating functions for partitions of numbers are strongly related to modular functions. This article introduces such connections using the Rogers-Ramanujan functions as key players. After exemplifying basic notions of partition theory and modular functions in tutorial manner, relations of modular functions to q-holonomic functions and sequences are discussed. Special emphasis is put on supplementing the ideas presented with concrete computer algebra. Despite intended as a tutorial, owing to the algorithmic focus the presentation might contain aspects of interest also to the expert. One major application concerns an algorithmic derivation of Felix Klein’s classical icosahedral equation.

Pure and Applied Mathematics Quarterly, 2008

Journal of Number Theory, 2016

In this paper, we establish two modular relations for the Rogers-Ramanujan-Slater functions of order fifteen. These relations are analogues to Ramanujan's famous forty identities for the Rogers-Ramanujan functions. Furthermore, we give interesting partition theoretic interpretations of these relations.

Tbilisi Mathematical Journal, 2014

In this paper, we establish some new modular equations of degree 9. We also establish several new P-Q mixed modular equations involving theta-functions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish some new general formulas for explicit evaluations of a Remarkable product of theta-functions.

1997

We start with a brief overview of the necessary theory: Given any cusp form f=∑ n≥ 1 an (f) qn of weight k, we denote by L (f, s) the L-function of f. For Re (s)> k/2+ 1, the value of L (f, s) is given by L (f, s)=∑ n≥ 1 an (f) ns and, one can show that L (f, s) has analytic continuation to the entire complex plane. The value of L (f, s) at s= k/2 will be of particular interest to us, and we will refer to this value as the central critical value of L (f, s).

In a manuscript of Ramanujan, published with his Lost Notebook there are forty identities involving the Rogers-Ramanujan functions. In this paper, we establish modular relations involving the Rogers-Ramanujan functions, the Rogers-Ramanujan type functions of order ten and the Rogers-Ramanujan-Slater type functions of order fifteen which are analogues to Ramanujan forty identities. We also give partition theoretic interpretations of our modular relations. 1977, D. Bressoud [12] in his doctoral thesis, proved 15 more from the list of 40. In 1989, A. F. J. Biagioli [9] proved 8 of the remaining 9 identities by invoking the theory of modular forms. Recently, B. C. Berndt et al. [8] have found new proofs for 35 of the forty identities in the spirit of Ramanujan. S. -S. Huang [16] and S. -L. Chen and Huang [13] have established several modular relations for the Göllnitz-Gordan functions by techniques which have been used by Rogers, Watson and Bressoud to prove some of Ramanujan's 40 identities. In 2008, N. D. Baruah, J. Bora and N. Saikia [6] have given alternative proofs some of them by using Schröter's formulas and some simple theta functions identities of Ramanujan. In 2003, H. Hahn [15] has established several modular relations for the septic analogues of the Rogers-Ramanujan functions. In 2007, Baruah and Bora [5] have established several modular relations for the nonic analogues of the Rogers-Ramanujan functions as well as relations that are connected with the Rogers-Ramanujan, Göllnitz-Gordan and septic analogues of Rogers-Ramanujan type functions. In 2007, Baruah and Bora [4] have established several modular relations involving two functions analogues to the Rogers-Ramanujan functions. Some of these relations are connected with Rogers-Ramanujan,

Journal of Mathematical Analysis and Applications, 2007

We study the linear relations among the Fourier coefficients of modular forms on the group Γ 0 (N) of genus zero. Applying these linear relations, congruence properties of Hecke eigenforms, replicable properties of Hauptmoduln and congruences of representation numbers of the sums of n squares can be obtained. The eta-quotient expression of the unique normalized modular form Δ N (z) of weight 12 on Γ 0 (N) with a zero of maximum order at ∞ is listed.

Proceedings of the American Mathematical Society, 2012

The authors have conjectured ([KoM]) that if a normalized generalized modular function (GMF) f , defined on a congruence subgroup Γ, has integral Fourier coefficients, then f is classical in the sense that some power f m is a modular function on Γ. A strengthened form of this conjecture was proved (loc cit) in case the divisor of f is empty. In the present paper we study the canonical decomposition of a normalized parabolic GMF f = f 1 f 0 into a product of normalized parabolic GMFs f 1 , f 0 such that f 1 has unitary character and f 0 has empty divisor. We show that the strengthened form of the conjecture holds if the first "few" Fourier coefficients of f 1 are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either f 0 = 1 or if the divisor of f is concentrated at the cusps of Γ.