### 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 | |

State | Published - Mar 20 2012 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*229*(5), 2603-2621. https://doi.org/10.1016/j.aim.2012.01.012

**The leading root of the partial theta function.** / Sokal, Alan D.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 229, no. 5, pp. 2603-2621. https://doi.org/10.1016/j.aim.2012.01.012

}

TY - JOUR

T1 - The leading root of the partial theta function

AU - Sokal, Alan D.

PY - 2012/3/20

Y1 - 2012/3/20

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

UR - http://www.scopus.com/inward/record.url?scp=84856508098&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856508098&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2012.01.012

DO - 10.1016/j.aim.2012.01.012

M3 - Article

VL - 229

SP - 2603

EP - 2621

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 5

ER -