Today's seminar
Seminar information archive ~02/12|Today's seminar 02/13 | Future seminars 02/14~
2025/02/13
Infinite Analysis Seminar Tokyo
13:30-17:00 Room #118, hybrid seminar (Graduate School of Math. Sci. Bldg.)
Genki Shibukawa (Kobe University) 13:30-15:00
Basics and developments of the $\mu$-function (First part) (日本語)
Basics and developments of the $\mu$-function (Second part) (日本語)
Genki Shibukawa (Kobe University) 13:30-15:00
Basics and developments of the $\mu$-function (First part) (日本語)
[ Abstract ]
First part (speaker: Genki Shibukawa) Starting from a short introduction to the original Zwegers' $\mu$ function, we explain some basic tools of $q$-special functions ( $q$-hypergeometric functions) and $q$-analysis ($q$-difference equations, $q$-Borel and $q$-Laplace transformations) which are necessary to give a special function theoretic interpretation of the $\mu $-function. In these settings, we introduce the generalized $\mu $-function which is a one-parameter deformation of the Zwegers' $\mu $-function, and derive its basic properties such as symmetry, explicit formulas, difference relations, and connection formulas. In particular, we explain the relationship between the generalized $\mu $-function and $q$-Hermite polynomial (function).
Satoshi Tsuchimi (Kobe University) 15:30-17:00First part (speaker: Genki Shibukawa) Starting from a short introduction to the original Zwegers' $\mu$ function, we explain some basic tools of $q$-special functions ( $q$-hypergeometric functions) and $q$-analysis ($q$-difference equations, $q$-Borel and $q$-Laplace transformations) which are necessary to give a special function theoretic interpretation of the $\mu $-function. In these settings, we introduce the generalized $\mu $-function which is a one-parameter deformation of the Zwegers' $\mu $-function, and derive its basic properties such as symmetry, explicit formulas, difference relations, and connection formulas. In particular, we explain the relationship between the generalized $\mu $-function and $q$-Hermite polynomial (function).
Basics and developments of the $\mu$-function (Second part) (日本語)
[ Abstract ]
Second part (speaker: Satoshi Tsuchimi)
In this part, we present some advanced topics of the $\mu$ function. First, we show that the $\mu$-functions naturally appear in special solutions of factorized higher-order $q$-difference equations. Next, by applying the above $q$-analytic methods, we introduce a multivariate analogue of the generalized $\mu$-function and give some formulas. Finally, by similar methods, we construct another generalization of the $\mu$-function from the Kontsevich function which is an important function in knot invariants. This generalization of the $\mu$-function is related to the big $q$-Hermite polynomial (function) which is a 1 parameter deform of the $q$-Hermite polynomial.
Second part (speaker: Satoshi Tsuchimi)
In this part, we present some advanced topics of the $\mu$ function. First, we show that the $\mu$-functions naturally appear in special solutions of factorized higher-order $q$-difference equations. Next, by applying the above $q$-analytic methods, we introduce a multivariate analogue of the generalized $\mu$-function and give some formulas. Finally, by similar methods, we construct another generalization of the $\mu$-function from the Kontsevich function which is an important function in knot invariants. This generalization of the $\mu$-function is related to the big $q$-Hermite polynomial (function) which is a 1 parameter deform of the $q$-Hermite polynomial.