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  2. Ramanujan–Sato series - Wikipedia

    en.wikipedia.org/wiki/Ramanujan–Sato_series

    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.

  3. Ramanujan–Petersson conjecture - Wikipedia

    en.wikipedia.org/wiki/Ramanujan–Petersson...

    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.

  4. Mock modular form - Wikipedia

    en.wikipedia.org/wiki/Mock_modular_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 .

  5. Ramanujan's congruences - Wikipedia

    en.wikipedia.org/wiki/Ramanujan's_congruences

    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".

  6. Modular form - Wikipedia

    en.wikipedia.org/wiki/Modular_form

    In mathematics, a modular form is a (complex) analytic function on the upper half-plane, , that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory.

  7. Cusp form - Wikipedia

    en.wikipedia.org/wiki/Cusp_form

    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 ...

  8. Ramanujan tau function - Wikipedia

    en.wikipedia.org/wiki/Ramanujan_tau_function

    Ramanujan (1916) observed, but did not prove, the following three properties of τ(n): τ(mn) = τ(m)τ(n) if gcd(m,n) = 1 (meaning that τ(n) is a multiplicative function); τ(p r + 1) = τ(p)τ(p r) − p 11 τ(p r − 1) for p prime and r > 0.

  9. Rogers–Ramanujan identities - Wikipedia

    en.wikipedia.org/wiki/Rogers–Ramanujan_identities

    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::