Eric T. Mortenson's research while affiliated with Saint Petersburg State University and other places
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Publications (30)
Using Appell function properties we give short proofs of Ramanujan-like identities for the eighth order mock theta function $V_0(q)$ after work of Chan and Mao; Mao; and Brietzke, da Silva, and Sellars. We also present a generalization of the identities in the spirit of celebrated results of Bringmann, Ono, and Rhoades on Dyson's ranks and Maass fo...
Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews--Crandall-type identity and use it to count the number of integer solutions to $x^2+2y^2+2z^2=n$.
We demonstrate how formulas that express Hecke-type double-sums in terms of theta functions and Appell–Lerch functions—the building blocks of Ramanujan’s mock theta functions—can be used to give general string function formulas for the affine Lie algebra A1(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfon...
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 p...
String functions are important building blocks of characters of integrable highest modules over affine Kac–Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type A1(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{...
We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent [Formula: see text]-hypergeometric function of Andrews as a mixed mock modular form.
In recent work where Matsusaka generalizes the relationship between Habiro-type series and false theta functions after Hikami, five families of Hecke-type double-sums of the form \begin{equation*} \left( \sum_{r,s\ge 0 }-\sum_{r,s<0}\right)(-1)^{r+s}x^ry^sq^{a\binom{r}{2}+brs+c\binom{s}{2}}, \end{equation*} where $b^2-ac<0$, are decomposed into sum...
Ramanujan'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 over eight sixth-ord...
We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent $q$-hypergeometric function of Andrews as a mixed mock modular form.
We express a family of Hecke--Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. Hikami and Lovejoy's family of Hecke--Appell-type sums was obtained by considering certain duals of generalized quantum modular forms.
We demonstrate how formulas that express Hecke-type double-sums in terms of theta functions and Appell--Lerch functions -- the building blocks of Ramanujan's mock theta functions -- can be used to give general string function formulas for the affine Lie algebra $A_{1}^{(1)}$ for levels $N=1,2,3,4$.
String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certa...
We develop a setting in which one can evaluate certain Hecke-Rogers series in terms of false theta functions. We apply our setting to recent false theta function identities of Chan and Kim as well as Andrews and Warnaar.
International audience
Using a heuristic that relates Appell-Lerch functions to divergent partial theta functions one can expand Hecke-type double-sums in terms of Appell-Lerch functions. We give examples where the heuristic can be used as a guide to evaluate analogous triple-sums in terms of Appell-Lerch functions or false theta functions.
Using a heuristic that relates Appell--Lerch functions to divergent partial theta functions one can expand Hecke-type double-sums in terms of Appell--Lerch functions. We give examples where the heuristic can be used as a guide to evaluate analogous triple-sums in terms of Appell--Lerch functions or false theta functions.
We develop a setting in which one can evaluate certain Hecke-Rogers series in terms of false theta functions. We apply our setting to recent false theta function identities of Chan and Kim as well as Andrews and Warnaar.
Denote by $p(n)$ the number of partitions of $n$ and by $N(a,M;n)$ the number of partitions of $n$ with rank congruent to $a$ modulo $M$. By considering the deviation \begin{equation*} D(a,M) := \sum_{n= 0}^{\infty}\left(N(a,M;n) - \frac{p(n)}{M}\right) q^n, \end{equation*} we give new proofs of recent results of Andrews, Berndt, Chan, Kim and Mali...
By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kronecker-type identity, we also deduce Gauss' theorem that every positive...
By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares.
Denote by p(n) the number of partitions of n and by N(a, M; n) the number of partitions of n with rank congruent to a modulo M. We find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average: $$\begin{aligned} D(a,M) := \sum _{n= 0}^{\infty }\left( N(a,M;n) - \frac{p(n)}{M}\right) q^n. \end{aligned}...
Using properties of Appell-Lerch functions, we give insightful proofs for six of Ramanujan's identities for the tenth-order mock theta functions.
We prove a double-sum analog of an identity known to Kronecker and then
express it in terms of functions studied by Appell and Kronecker's student
Lerch, in so doing we show that the double-sum analog is of mixed mock modular
form.
Using results from Ramanujan’s lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom
et al.
for mock theta functions. Here we see that Mortenson’s previous work on the dual nature of Appell–Lerch sums and partial theta functions and on constructing bilateral
q
-series with mixed mock modular behaviour is well s...
In recent work, Bringmann, Ono, and Rhoades employ the theory of harmonic
weak Maass forms to prove results on Eulerian series as modular forms. By
changing the setting to Appell--Lerch sums, after recent work of Hickerson and
the author, we shorten the proof of one of the main theorems of Bringmann {\it
et al.} and also clarify speculation of Soon...
In recent work, Hickerson and the author demonstrated that it is useful to
think of Appell-Lerch sums as partial theta functions. This notion can be used
to relate identities involving partial theta functions with identities
involving Appell-Lerch sums. In this sense, Appell-Lerch sums and partial theta
functions appear to be dual to each other. We...
Kaplansky [2003] proved a theorem on the simultaneous representation of a
prime $p$ by two different principal binary quadratic forms. Later, Brink found
five more like theorems and claimed that there were no others. By putting
Kaplansky-like theorems into the context of threefield identities after
Andrews, Dyson, and Hickerson, we find that there...
By developing a connection between partial theta functions and Appell-Lerch
sums, we find and prove a formula which expresses Hecke-type double sums in
terms of Appell-Lerch sums and theta functions. Not only does our formula prove
classical Hecke-type double sum identities such as those found in work Kac and
Peterson on affine Lie Algebras and Hec...
We use a specialization of Ramanujan’s 1ψ
1 summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell–Lerch sums and theta functions.
We obtain four Hecke-type double sums for three of Ramanujan’s third order mock theta functions. We discuss how these four are related to the new mock theta functions of G. E. Andrews ’ work on q-orthogonal polynomials [Proc. Steklov Inst. Math. 276, No. 1, 21–32 (2012); translation from Teoriya Chisel, Algebra i Analiz, Tr. Mat. Inst. Steklova 276...
Citations
... In [6,16], double-sums of the form (1.2) where b 2 − ac > 0 are extensively studied. Expansions are obtained that express the double-sums in terms of Appell functions, i.e. the building blocks of Ramanujan's mock theta functions, and theta functions. ...
... Proof of Proposition 7.1 Identity (7.1a) is true by [8,Lemma 3.11]. We prove (7.1b). ...
... This setting contrasts with [3,15], where false theta functions are expressed in terms of doublesums where instead there is a plus sign between the summation symbols in (1.2) and we do not have the restriction b 2 − ac < 0. It also contrasts with the setting where b 2 − ac > 0. ...
... Kang [13] deduced the generating functions of two variations of three combination of ranks and then derived the generating function of the combination of ranks modulo 6. Aygin and Chan [3] gave the generating functions of two variations of three combinations of cranks and the generating functions of the combinations of cranks modulo 6, 9, and 12. For more identities on ranks and cranks, see [5,6,[10][11][12][14][15][16]. ...
Reference: On cranks of partitions modulo 12 and 16
... 10.6.1]. The case r = 3 of (1.12) gives class numbers of imaginary quadratic fields; see [32], for instance, for the connection between N 3 (n) and class numbers. ...
... Zwegers [21] has since given short proof of identities (1.1)-(1.4). Mortenson [16] later gave short proofs of all six identities (1.1)- (1.6). ...
... . [13,Section 8]. Determining the theta function is a difficult task. ...
... Determining the theta function is a difficult task. Sometimes, one can obtain the theta function in the course of a direct proof [11,16]. ...
... For more details on precisely how this works, along with many examples, see [21]. The double sums like the one on the right-hand side of (1.11) lead naturally to instance of the theta function When ac < b 2 these are indefinite theta functions which are intimately related to mock theta functions [14,29], and so it is not surprising that the conjugate Bailey pairs arising from residual identities have been useful in a number of studies [17][18][19]22]. Recently, a generalization of (the case a = 1 of) (1.11) was proved by Hikami and the second author using a multisum residual identity of Warnaar [28]. ...
... In short, for a prime t ≥ 5, 0 ≤ r 1 , r 2 < t and 0 ≤ d < t, they found conditions such that ∞ n=0 (N(r 1 , t; tn + d) − N(r 2 , t; tn + d)) q 24(tn+d)−1 is a weight 1/2 weakly holomorphic modular form on the congruence subgroup Γ 1 (576t 6 ) [4, Theorem 1.1]. One also finds an approach to this result using new Appell function properties [10, Section 3] in [11]. ...
