Hypergeometric Forms for Ising-Class Integrals

Abstract

We apply experimental-mathematical principles to analyze integralsThese are generalizations of a previous integral C n := C n,1 relevant to the Ising theory of solid-state physics [8]. We find representations of the C n,k in terms of Meijer G-functions and nested-Barnes integrals. Our investigations began by computing 500-digit numerical values of C n,k for all integers n, k where n ∈ [2, 12] and k ∈ [0, 25]. We found that some C n,k enjoy exact evaluations involving Dirichlet L-functions or the Riemann zeta function. In the process of analyzing hypergeometric representations, we found-experimentally and strikingly-that the C n,k almost certainly satisfy certain inter-indicial relations including discrete k-recursions. Using generating functions, differential theory, complex analysis, and Wilf-Zeilberger algorithms we are able to prove some central cases of these relations.

High-precision computation: Mathematical physics and dynamics

“…The conjectured identity shown here for E 5 was confirmed to 240-digit accuracy, which is 180 digits beyond the level that could reasonably be ascribed to numerical round-off error; thus we are quite confident in this result even though we do not have a formal proof [6]. In a follow-on study [8], we examined the following generalization of the C n integrals:…”

“…Here we made the initially surprising discovery-now proven in [30]-that there are linear relations in each of the rows of this array (considered as a doubly-infinite rectangular matrix), e.g., 0 = C 3,0 − 84C 3,2 + 216C 3,4 0 = 2C 3,1 − 69C 3,3 + 135C 3,5 0 = C 3,2 − 24C 3,4 + 40C 3,6 0 = 32C 3,3 − 630C 3,5 + 945C 3,7 0 = 125C 3,4 − 2172C 3,6 + 3024C 3,8 .…”

“…But whether or not library software is used, a double-precision implementation of this algorithm fails to find the correct underlying polynomial coefficients. However, an implementation of this scheme using "double-double" precision (i.e., roughly 31-digit precision) correctly deduces that the original data sequence is given by the polynomial function f (k) = 5 + 220k 2 + 990k 4 + 924k 6 + 165k 8 .…”

“…Suppose we wish to fit the following data to a polynomial: 5, 2304, 118101, 1838336, 14855109, 79514880, 321537749, 1062287616, 3014530821, for integer arguments (0, ..., 8). The usual approach is to employ polynomial least squares curve fitting, which amounts to solving a (n + 1 × n + 1) linear system of equations (written in matrix form): In a computer implementation of this algorithm, the linear equation solution is most commonly (and most wisely) done with library software, such as the Linpack [48] or LAPACK [47].…”

The Sunrise Integral Around Two and Four Space-time Dimensions in Terms of Elliptic Polylogarithms

“…Because of its fundamental character the sunrise integral received numerous attention in the past [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. For the equal mass case an analytic result involving elliptic integrals and elliptic polylogarithms was provided in [10,16].…”