Theta Function Identities Research Articles

Proof of a conjecture on congruences of the sum of minimal excludants

Very recently, Baruah et al. refined a result of Andrews and Newman on the sum of minimal excludants over all the partitions of a positive integer [Formula: see text] by establishing the generating functions of two functions [Formula: see text] and [Formula: see text] which denote the sum of odd minimal excludants and the sum of even minimal excludants, respectively. They proved some congruences on [Formula: see text] and [Formula: see text] modulo 4 and 8 and posed a conjecture on [Formula: see text] and [Formula: see text] at the end of their paper. In this paper, we confirm this conjecture by using theta function identities.

Probabilistic aspects of Jacobi theta functions

In this note we deduce well-known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

Proofs of three conjectured internal congruences modulo 32 for Schur-type overpartitions

Let S(n) denote the number of overpartitions of Schur-type. Recently, Chern, da Silva and Sellers proved many congruences modulo 8 and 16 for S(n). At the end of their paper, they also posed a conjecture on internal congruences modulo 32 for S(n). In this paper, we confirm their conjecture by using theta function identities and the (p, k)-parameterization of theta functions given by Alaca and Williams. In particular, we show that for any integer j with 0 ≤ j ≤ 31, there are infinitely many integers uj such that S(uj ) ≡ j (mod 32).

On certain relations among the generating functions for certain quadratic forms

The object of this article is to establish the relation between the generating function of the quadratic form $2m^2+2mn+3n^2$ {and} the generating function{s} for the quadratic forms $m^2+mn+n^2$, $m^2+mn+2n^2$, $m^2+mn+4n^2$ and $2m^2+mn+2n^2$. In the process, we deduce certain interesting theta function identities.

Some congruences for 12-coloured generalized Frobenius partitions

AbstractIn his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$. In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$.

Infinite families of congruences modulo $2$ for $(\ell, k)$-regular partitions

Let $b_{\ell, k}(n)$ denote the number of $(\ell, k)$-regular partition of $n$. Recently, some congruences modulo $2$ for $ (3, 8), (4, 7)$-regular partition and modulo $8$, modulo $9$ and modulo $12$ for $(4, 9)$-regular partition has been studied. In this paper, we use theta function identities and Newman results to prove some infinite families of congruences modulo $2$ for $(2, 7)$, $(5, 8)$, $(4, 11)$-regular partition and modulo $4$ for $(4, 5)$-regular partition.

On Ramanujan's lost notebook and new tenth‐order like identities for second‐, sixth‐, and eighth‐order mock theta functions

AbstractRamanujan's lost notebook contains many mock theta functions and mock theta function identities not mentioned in his last letter to Hardy. 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. We also find eight sixth‐order mock theta functions in the lost notebook, but among their many identities there is only a single relationship like those of the tenth‐orders. Using Appell function properties of Hickerson and Mortenson, we discover and prove three new identities for the sixth‐order mock theta functions that are in the spirit of the six tenth‐order identities. We also include an additional nineteen tenth‐order like identities for various combinations of second‐, sixth‐, and eighth‐order mock theta functions.

On Ramanujan’s continued fractions of order twenty-four

Two continued fractions U(q) and V(q) of order twenty-four are obtained from a general continued fraction identity of Ramanujan. Some theta-function and modular identities for U(q) and V(q) are established to prove general theorems for the explicit evaluations of U(±q) and V(±q). From the theta-function identities of U(q) and V(q), three colour partition identities are derived as application to partition theory of integer. Further, 2-, 4- and 8-dissection formulas are established for the continued fractions U∗(q)≔q−5/2U(q) and V∗(q)≔q−1/2V(q), and their reciprocals.

ON ROGERS-RAMANUJAN-SLATER TYPE THETA FUNCTION IDENTITY

The main purpose of this article is to prove two Rogers–Ramanujan–Slater type theta function identities related to φ(q) and ϕ0(q), which were earlier investigated by two legendary mathematicians of their time. The results presented in this paper are motivated essentially by recent works of Cao et al. (see [5]).

The three missing terms in Ramanujan’s septic theta function identity

On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function varphi (q): φ(e-7π7)=7-3/4φ(e-π7){1+()2/7+()2/7+()2/7}.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\varphi (e^{-7\\pi \\sqrt{7}}) = 7^{-3/4}\\varphi (e^{-\\pi \\sqrt{7}})\\big \\{1 + (\\quad )^{2/7} + (\\quad )^{2/7} + (\\quad )^{2/7}\\big \\}. \\end{aligned}$$\\end{document}We give the three missing terms. In addition, we calculate the class invariant G_{343} and further special values of varphi (e^{-npi }), for n = 7, 21, 35, and 49.

New proofs for three identities of seventh order mock theta functions

<abstract><p>Using the three-term Weierstrass relation for theta functions and the properties of Hecke-type double sums and Appell-Lerch sums, we give new and simple proofs for the seventh order mock theta conjectures.</p></abstract>

A NOTE ON ROGERS-RAMANUJAN-SLATER TYPE THETA FUNCTION IDENTITY

In this paper, we research theta function identity involving Rogers– Ramanujan identity and establish a Rogers–Ramanujan–Slater type theta function identity related to G(q) and φ(q).

Jacobi’s Triple-Product Identity and an Associated Family of Theta-Function Identities

Jacobi’s Triple-Product Identity and an Associated Family of Theta-Function Identities

Eisenstein series of level 6 and level 10 with their applications to theta function identities of Ramanujan

S. Ramanujan recorded theta function identities of different levels in the unorganized pages of his second notebook and the lost notebook. In this paper, we prove level 6 and level 10 theta function identities by using Eisenstein series identities.

On two theta function identities of Ramanujan

On two theta function identities of Ramanujan

Products of two 1ϕ1 series and implications

An expansion formula into bivariate formal power series is established that implies, as consequences, three remarkable identities of partial theta functions due to Warnaar (2003), Schilling–Warnaar (2002) and Andrews–Warnaar (2007).

Proofs of some conjectures of Sun on the relations between T(a,b,c,d;n) and N(a,b,c,d;n)

Let N(a,b,c,d;n) be the number of representations of n as ax2+by2+cz2+dw2 and T(a,b,c,d;n) be the number of representations of n as aX(X+1)2+bY(Y+1)2+cZ(Z+1)2+dW(W+1)2, where a,b,c,d are positive integers, x,y,z,w are integers, and n,X,Y,Z,W are nonnegative integers. In this paper, by using theta function identities, we confirm Sun’s conjectures that T(a,b,c,d;n) is a linear combination of N(a,b,c,d;m) and N(a,b,c,d;4m), with m=8n+a+b+c+d, for several values of a,b,c,d.

On Gosper’s $\Pi_q$ and Lambert series identities

In an interesting article entitled “Experiments and discoveries in $q$-trigonometry”, R. W. Gosper conjectured few beautiful $\Pi_q$ and Lambert series identities. Many people have attempted confirming some of those identities in the Gosper’s list, mainly by using Gosper’s $q$-trigonometric identities. In this paper we either prove or disprove all the $\Pi_q$ and Lambert series identities in the Gosper’s list by mainly using S. Ramanujan’s theta function identities and W. N. Bailey’s summation formula. In the process, we obtain three new Gosper kind of identities.

On Ramanujan's Eisenstein series identities

Ramanujan's work was inspiring and distinguishable from those of his predecessors as he wrote many theorems, but he proved very few. In this paper, we make use of P-Q theta function identities written by Ramanujan and we offer new proofs to Eisenstein series identities of level 14 developed by Cooper and Ye, where the proofs were presented using modular forms. Analogously, we derive certain new Eisenstein series identities of level 14.

Jacobi's Triple-Product Identity and an Associated Family of Theta-Function Identities

The main object of this paper is to present a family of $q$-series identities which involve some of the theta functions of Jacobi and Ramanujan. Each of these (presumably new) $q$-series identities reveals interesting relationships among three of the theta-type functions which stem from the celebrated Jacobi's triple-product identity in a remarkably simple way. The results presented in this paper are motivated essentially by a number of recent works dealing with the subject matter which is investigated herein.