The leading root of the partial theta function

  title={The leading root of the partial theta function},
  author={Alan D. Sokal},
  journal={Advances in Mathematics},
  • A. Sokal
  • Published 6 June 2011
  • Mathematics
  • Advances in Mathematics

On a partial theta function and its spectrum

  • V. Kostov
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2016
The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑j=0∞qj(j+1)/2xj, where x ∈ ℂ is a variable and q ∈ ℂ, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the

2 0 N ov 2 01 9 Two properties of the partial theta function

For the partial theta function θ(q, z) := ∑ ∞ j=0 q z, q, z ∈ C, |q| < 1, we prove that its zero set is connected. This set is smooth at every point (q, z) such that z is a simple or double zero of

On the Complex Conjugate Zeros of the Partial Theta Function

  • V. P. Rostov
  • Mathematics, Philosophy
    Functional Analysis and Its Applications
  • 2019
We prove that (1) for any q ∈ (0, 1), all complex conjugate pairs of zeros of the partial theta function $$\theta (q,x): = \sum\nolimits_{j = 0}^\infty {{q^{j(j +

On the entire functions from the Laguerre–Pólya class having the decreasing second quotients of Taylor coefficients

No zeros of the partial theta function in the unit disk

We prove that for q ∈ ( − 1 , 0) ∪ (0 , 1), the partial theta function θ ( q, x ) := P ∞ j =0 q j ( j +1) / 2 x j has no zeros in the closed unit disk.

On the solution to a certain functional differential equation

which is the unique smooth solution to the functional differential equation(see[2]) f (x) = f(qx), 0 < q < 1, f(0) = 1. It was proved by Morris et al.[5] that f(x) has infinitely many real roots and


In 1907, M. Petrovitch initiated the study of a class of entire functions all whose finite sections (i.e., truncations) are real-rooted polynomials. He was motivated by previous studies of E.

A Separation in Modulus Property of the Zeros of a Partial Theta Function

We consider the partial theta function $$\theta (q,z): = \sum\nolimits_{j = 0} \infty {{q {j(j + 1)/2}}{z j}} $$θ(q,z):=∑j=0∞qj(j+1)/2zj, where z ∈ ℂ is a variable and q ∈ ℂ, 0 < |q| < 1, is a

Stabilization of the asymptotic expansions of the zeros of a partial theta function

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. We prove a property of



On Some Conjectures by Morriset al.about Zeros of an Entire Function

Abstract It is known that the entire function ∑ ∞ n  = 0 (− z ) n q n ( n  − 1)/2 / n !, q  ∈ (0, 1), has infinitely many positive but no other zeros in the complex plane. A number of conjectures on

A basic hypergeometric transformation of Ramanujan and a generalization

The q-identity (−bq)∞∑∞n=0qn(n+1)/2(−λ/a)nan(q)n(−bq)n=(−aq)∞∑∞n=0qn(n+1)/2(−λ/b)nbn(q)n(−aq)n is established by considering some simple functional relations of its members. This gives a

Partial Theta Functions. I. Beyond the Lost Notebook

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and


Abstract We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special

Some canonical sequences of integers


In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was

A combinatorial study and comparison of partial theta identities of Andrews and Ramanujan

We provide a simple proof of a partial theta identity of Andrews and study the underlying combinatorics. This yields a weighted partition theorem involving partitions into distinct parts with


In this brief note, we give combinatorial proofs of two identities involving partial theta functions. As an application, we prove an identity for the product of partial theta functions, first

The theory of partitions

1. The elementary theory of partitions 2. Infinite series generating functions 3. Restricted partitions and permutations 4. Compositions and Simon Newcomb's problem 5. The Hardy-Ramanujan-Rademacher