We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series an...
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We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer n by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form.
We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these mer...
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We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke's indefinite theta functions.
Two classes of finite trigonometric sums, each involving only sines, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations.
Two classes of finite trigonometric sums, each involving only sines, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations.
Ramanujan's last letter to Hardy introduced the world to mock theta functions, and the mock theta function identities found in Ramanujan's lost notebook added to their intriguing nature. For example, we find t...
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Ramanujan's last letter to Hardy introduced the world to mock theta functions, and the mock theta function identities found in Ramanujan's lost notebook added to their intriguing nature. For example, we find the four tenth-order mock theta functions and their six identities. The six identities themselves are of a spectacular nature and were first proved by Choi. Indeed, in their fifth and final volume on Ramanujan's lost notebook, Andrews and Berndt proclaimed about one of the six identities "It is inconceivable that an identity such as J1,2j(-q2;-q10) phi 10(q) - q-1 psi 10(-q4) + q-2 chi 10(q8) = J2,8 could be stumbled upon by a mindless search algorithm without any overarching theoretical insight." Recently, the first author discovered and proved three new tenth-order like identities but for sixth-order mock theta functions. Building on this recent discovery, we develop a long sought after overarching theoretical insight hinted at by Andrews and Berndt by finding several general families of tenth-order like identities for Appell functions, which are the building blocks of Ramanujan's mock theta functions. We underscore how this adds to the mystery of the purported missing pages of Ramanujan's lost notebook. (c) 2024 Elsevier Inc. All rights reserved.
We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transforma...
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We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Z(n) whose initial condition is given by a linear code.
Let F be a totally real number field and o the ring of integers of F. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group SL2(o). We obtain an equivalent cond...
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Let F be a totally real number field and o the ring of integers of F. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group SL2(o). We obtain an equivalent condition that there exists a multiplier system of half-integral weight for SL2(o). We determine the condition of F that there exists a theta function which is a Hilbert modular form of half-integral weight for SL2(o). The theta function is defined by a sum on a fractional ideal of F. (c) 2022 Elsevier Inc. All rights reserved.
We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type I. In our framework, each theta relation is controlled by combinatorial properties ...
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We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type I. In our framework, each theta relation is controlled by combinatorial properties of a pair (T, P) of a regular matrix T over an imaginary quadratic field and a positive-definite Hermitian matrix P over the complex number field.& COPY;2023 Elsevier Inc. All rights reserved.
The study on vanishing coefficients with arithmetic progressions in quotients of theta functions has its origin in the work by Richmond and Szekeres. Further investigations were subsequently considered by Andrews and ...
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The study on vanishing coefficients with arithmetic progressions in quotients of theta functions has its origin in the work by Richmond and Szekeres. Further investigations were subsequently considered by Andrews and Bressoud, Alladi and Gordon, and Mc Laughlin. Quite recently, Du and the author found that certain arithmetic progressions on vanishing coefficients could be enjoyed by a family of quotients of theta functions. In this paper, we prove that there are more such instances in two families of quotients of theta functions. Finally, we present several related conjectures that merit further investigation.
作者：
Chen, Shi-ChaoHenan Univ
Inst Contemporary Math Dept Math & Stat Sci Kaifeng Peoples R China
We establish some vanishing results on the coefficients of products of one variable theta functions. As applications, we prove some congruence properties of Andrews' partition function (eO) over bar(n) and refine ...
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We establish some vanishing results on the coefficients of products of one variable theta functions. As applications, we prove some congruence properties of Andrews' partition function (eO) over bar(n) and refine Ray and Barman's congruences.
Following recent studies of vanishing coefficients in arithmetic progressions in infinite q-series products, we further prove that such phenomenon also exists in a family of quotients of theta functions with odd modul...
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Following recent studies of vanishing coefficients in arithmetic progressions in infinite q-series products, we further prove that such phenomenon also exists in a family of quotients of theta functions with odd moduli. Moreover, we find that some arithmetic progressions with vanishing coefficients could be shared by an infinite family of quotients of theta functions. Finally, we present several related conjectures that merit further investigation.
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