Multiple gamma functions, multiple sine functions, and Appell’s O-functions

Abstract

Kurokawa introduced q-multiple gamma functions and q-multiple sine functions. We show that the Appell's O-function is expressed via the q-multiple gamma function. We also give some applications of this result. For example, we obtain a formula for the "Stirling modular form" and calculate special values of the q-multiple sine function. Moreover, we give some formulas of Eisenstein series and double cotangent functions and its generalization. Then the former gives an infinite product expression of the double sine function explicitly and a result of Kurokawa.

Spectral analysis of stable processes on the positive half-line

“…Thus we have verified that that the functions f and f satisfy (27) for z > 0, and the proof for z < 0 follows by taking the complex conjugate in the above equation.…”

“…This function appears in the definition (12) of the eigenfunctions, in formulas (19) and (20) that give the Laplace and Mellin transform of the eigenfunctions and in formula (26), which gives the Wiener-Hopf factors of stable processes. So far, apart from some number-theoretic applications [13,27], this curious special function has appeared mostly in the Physics literature [9,23,25,30], where it is used in studying quantum topology and cluster algebras. It is an interesting question whether this appearance of the double sine function in our study of stable processes is simply a coincidence or there is indeed a deeper connection between stable processes on the half-line and quantum topology and/or cluster algebras.…”

“…For fixed α ∈ C \ (−∞, 0], the double sine function is meromorphic in z-variable and it admits the product representation of the form where P (z) := (1 − z) exp(z + z 2 /2) and the prime in the second product means that the term corresponding to m = n = 0 is omitted. The explicit expressions for A(α), B(α) and C(α) can be found in [27,Theorem 6]. It is clear from (A.5) that S 2 (z; α) has zeros at points {−m − αn : m ≥ 0, n ≥ 0} and poles at points {m + αn : m ≥ 1, n ≥ 1}.…”

“…This is a classical Riemann-Hilbert problem: we need to find a function φ(−iz) analytic in the upper half-plane Im(z) > 0 and continuous in the closed upper half-plane and another functionφ(iz) having the same properties but in the lower half-plane Im(z) < 0 which meet at the boundary Im(z) = 0 as prescribed by (27). Let us define by f (z) the function in the right-hand side of (26), and byf (z) the same function, but with ρ replaced byρ.…”

“…Let us define by f (z) the function in the right-hand side of ( 26), and by f (z) the same function, but with ρ replaced by ρ. First we will verify that the functions f and f satisfy (27). Assume that z > 0 and let us denote w = α ln(z)/(2πi).…”