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The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent (k − 1)/2 where k is the weight of the form.
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1 / 2 . The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook .
In mathematics, a Ramanujan–Sato series [1] [2] generalizes Ramanujan’s pi formulas such as, = = ()!! + to the form = = + by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients (), and ,, employing modular forms of higher levels.
The Rogers–Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics. The demodularized standard form of the Ramanujan's continued fraction unanchored from the modular form is as follows::
The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS Regional Conference Series in Mathematics. Vol. 102. Providence, RI: American Mathematical Society. ISBN 978-0-8218-3368-1. Zbl 1119.11026. Ramanujan, S. (1919). "Some properties of p(n), the number of partitions of n".
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) ... a modular form of weight 1/2, as, [1] () ...
For example, the Ramanujan tau function τ(n) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with a 1 = 1. The space of such forms has dimension 1, which means this definition is possible; and that accounts for the action of Hecke operators on the space being by scalar multiplication (Mordell ...
Mordell () used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by Hecke (1937a,1937b).Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form,