A study of elliptic gamma function and allies

@article{Pasol2018ASO,
  title={A study of elliptic gamma function and allies},
  author={Vicentiu Pasol and Wadim Zudilin},
  journal={Research in the Mathematical Sciences},
  year={2018},
  volume={5},
  pages={1-11}
}
We study analytic and arithmetic properties of the elliptic gamma function $$\begin{aligned} \prod _{m,n=0}^\infty \frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \quad |q|,|p|<1, \end{aligned}$$∏m,n=0∞1-x-1qm+1pn+11-xqmpn,|q|,|p|<1,in the regime $$p=q$$p=q, in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eisenstein series and, via some formulae of D. Zagier, to elliptic… 
From multi-gravitons to Black holes: The role of complex saddles
By applying the Atiyah-Bott-Berline-Vergne equivariant integration formula upon double dimensional integrals, we find a way to compute the matrix integral representations of $4d$ $\mathcal{N}=1$
Introduction to the Theory of Elliptic Hypergeometric Integrals
We give a brief account of the key properties of elliptic hypergeometric integrals—a relatively recently discovered top class of transcendental special functions of hypergeometric type. In
The large-$N$ limit of 4d superconformal indices for general BPS charges
We study the superconformal index of N = 1 quiver theories at large -N for general values of electric charges and angular momenta, using both the Bethe Ansatz formulation and the more recent elliptic
Short Walk Adventures
We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend
Supersymmetric phases of 4d $$ \mathcal{N} $$ = 4 SYM at large N
We find a family of complex saddle-points at large N of the matrix model for the superconformal index of SU(N) N=4 super Yang-Mills theory on $S^3 \times S^1$ with one chemical potential $\tau$. The
The large-N limit of the 4d $$ \mathcal{N} $$ = 1 superconformal index
We systematically analyze the large-$N$ limit of the superconformal index of $\mathcal{N}=1$ superconformal theories having a quiver description. The index of these theories is known in terms of

References

SHOWING 1-9 OF 9 REFERENCES
The Elliptic Gamma Function and SL(3, Z)⋉Z3
Abstract The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma
Mahler Measure Variations, Eisenstein Series and Instanton Expansions
This paper points at an intriguing inverse function relation with on the one hand the coefficients of the Eisenstein series in Rodriguez Villegas’ paper on “Modular Mahler Measures” and on the
First order analytic difference equations and integrable quantum systems
We present a new solution method for a class of first order analytic difference equations. The method yields explicit “minimal” solutions that are essentially unique. Special difference equations
Multiplication Formulas for the Elliptic Gamma Function
The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove
On a formula of Bloch
We give a new proof of a formula of Bloch for a special value of a certain Eisenstein series of weight one with an additive character.
Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves
Introduction Tamagawa numbers Tamagawa numbers. Continued Continuous cohomology A theorem of Borel and its reformulation The regulator map. I The dilogarithm function The regulator map. II The
Mahler measure variations
  • Eisenstein series and instanton expansions, in “Mirror symmetry V”, AMS/IP Stud. Adv. Math. 38
  • 2006
Higher regulators
  • algebraic K-theory, and zeta functions of elliptic curves, Lecture notes (UC Irvine, 1977); CRM Monograph Ser. 11
  • 2000