The leading root of the partial theta function

Alan D. Sokal

doi:10.1016/j.aim.2012.01.012

The leading root of the partial theta function

Alan D. Sokal

Research output: Contribution to journal › Article › peer-review

Abstract

I study the leading root x ₀(y) of the partial theta function Θ0(x,y)=∑n=0∞xnyn(n-1)/2, considered as a formal power series. I prove that all the coefficients of -x ₀(y) are strictly positive. Indeed, I prove the stronger results that all the coefficients of -1/x ₀(y) after the constant term 1 are strictly negative, and all the coefficients of 1/x ₀(y) ² after the constant term 1 are strictly negative except for the vanishing coefficient of y ³.

Original language	English (US)
Pages (from-to)	2603-2621
Number of pages	19
Journal	Advances in Mathematics
Volume	229
Issue number	5
DOIs	https://doi.org/10.1016/j.aim.2012.01.012
State	Published - Mar 20 2012

Keywords

Formal power series
Implicit function theorem
Partial theta function
Q-Series
Rogers-Ramanujan function
Root

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2012.01.012

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title = "The leading root of the partial theta function",

abstract = "I study the leading root x 0(y) of the partial theta function Θ0(x,y)=∑n=0∞xnyn(n-1)/2, considered as a formal power series. I prove that all the coefficients of -x 0(y) are strictly positive. Indeed, I prove the stronger results that all the coefficients of -1/x 0(y) after the constant term 1 are strictly negative, and all the coefficients of 1/x 0(y) 2 after the constant term 1 are strictly negative except for the vanishing coefficient of y 3.",

keywords = "Formal power series, Implicit function theorem, Partial theta function, Q-Series, Rogers-Ramanujan function, Root",

author = "Sokal, {Alan D.}",

note = "Funding Information: This research was supported in part by U.S. National Science Foundation grant PHY– 0424082.",

year = "2012",

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N2 - I study the leading root x 0(y) of the partial theta function Θ0(x,y)=∑n=0∞xnyn(n-1)/2, considered as a formal power series. I prove that all the coefficients of -x 0(y) are strictly positive. Indeed, I prove the stronger results that all the coefficients of -1/x 0(y) after the constant term 1 are strictly negative, and all the coefficients of 1/x 0(y) 2 after the constant term 1 are strictly negative except for the vanishing coefficient of y 3.

AB - I study the leading root x 0(y) of the partial theta function Θ0(x,y)=∑n=0∞xnyn(n-1)/2, considered as a formal power series. I prove that all the coefficients of -x 0(y) are strictly positive. Indeed, I prove the stronger results that all the coefficients of -1/x 0(y) after the constant term 1 are strictly negative, and all the coefficients of 1/x 0(y) 2 after the constant term 1 are strictly negative except for the vanishing coefficient of y 3.

KW - Formal power series

KW - Implicit function theorem

KW - Partial theta function

KW - Q-Series

KW - Rogers-Ramanujan function

KW - Root

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JO - Advances in Mathematics

JF - Advances in Mathematics

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ER -

The leading root of the partial theta function

Abstract

Keywords

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