[Paper Review] Epsilon expansion of Appell and Kampé de Fériet functions

David Greynat, J. Sesma+1 more|arXiv (Cornell University)|Oct 29, 2013

Mathematical functions and polynomials3 references4 citations

TL;DR

This paper presents a systematic method for computing epsilon-expansions of Appell and Kampé de Fériet functions—key in Feynman diagram calculations—by deriving high-order derivatives of Pochhammer and reciprocal Pochhammer symbols. The approach leverages Taylor expansions in ε around zero, using closed-form expressions involving generalized harmonic numbers and Stirling numbers, enabling efficient, analytical computation of epsilon expansions for hypergeometric functions with parameters linear in ε.

ABSTRACT

The decomposition in partial fractions of the quotient of Pochhammer symbols improves considerably a method, suggested in a precedent paper, which allows one to obtain the $\varepsilon$-expansion of functions of the hypergeometric class. The procedure is applied to several Appell and Kampé de Fériet functions considered in the literature. Explicit expressions and interesting properties of the derivatives of the Pochhammer and reciprocal Pochhammer symbols, which are essential elements in the procedure, are given in an appendix.

Motivation & Objective

• To develop a general, efficient method for computing epsilon-expansions of hypergeometric functions with parameters linear in ε, particularly Appell and Kampé de Fériet functions.
• To derive closed-form expressions for the k-th order derivatives of Pochhammer and reciprocal Pochhammer symbols with respect to their arguments.
• To apply the method to compute full epsilon-expansions of specific functions arising in one-loop and two-loop Feynman diagrams.
• To introduce and utilize partial fraction decompositions of quotients of Pochhammer symbols to simplify higher-order derivative computations.

Proposed method

• Derives general formulas for the k-th order derivatives of the Pochhammer symbol (a)_{m} and its reciprocal using generalized harmonic numbers and Stirling numbers of the first kind.
• Introduces the operators τ_{m}^{(k)}(a) and ρ_{m}^{(k)}(b) to represent normalized k-th derivatives of (a)_{m} and 1/(b)_{m}, respectively.
• Applies Taylor expansion in ε around ε=0 to the hypergeometric functions, using the derived derivatives as building blocks.
• Employs partial fraction decomposition of quotients of Pochhammer symbols into simple poles to facilitate differentiation with respect to ε.
• Uses generating functions and combinatorial identities involving binomial coefficients and generalized harmonic numbers to express results in compact form.
• Validates the method by computing full ε-expansions for Appell and Kampé de Fériet functions from known physical diagrams, including pentagon and sunrise diagrams.

Experimental results

Research questions

• RQ1 How can high-order derivatives of Pochhammer and reciprocal Pochhammer symbols be systematically expressed in terms of known special functions?
• RQ2 What is the most efficient way to compute the ε-expansion of Appell and Kampé de Fériet functions with parameters linear in ε?
• RQ3 Can partial fraction decomposition of Pochhammer quotients simplify the derivation of ε-expansions in multi-variable hypergeometric functions?
• RQ4 What role do generalized harmonic numbers and Stirling numbers play in the analytical structure of ε-expansions?
• RQ5 How can the proposed method be applied to compute full ε-expansions for specific Feynman diagrams, such as the one-loop pentagon and two-loop sunrise diagram?

Key findings

• The k-th order derivative of the Pochhammer symbol (a)_{m} with respect to a is given by τ_{m}^{(k)}(a) = ∑_{j=0}^{m-1} Σ_{k}(j) / (a+j)^k, where Σ_{k}(j) involves generalized harmonic numbers.
• The reciprocal Pochhammer symbol 1/(b)_{m} has k-th derivative ρ_{m}^{(k)}(b) = (-1)^k ∑_{j=0}^{m-1} Σ_{k}(j) / (b+j)^k, with identical Σ_{k}(j) coefficients.
• The method successfully computes the full ε-expansion of the Appell F1 function in the one-loop pentagon diagram, matching known results.
• For the two-loop sunrise diagram, the method yields the complete ε-expansion of the relevant Kampé de Fériet function up to arbitrary order in ε.
• Partial fraction decomposition of quotients like (1+δ−ε)_{n1+n2}/[(1−ε)_{n1}(1+δ+ε)_{n2}] enables systematic derivation of ε-expansions via residue-like terms.
• The approach provides a unified, analytical framework for ε-expansions of hypergeometric functions in quantum field theory, avoiding numerical or series-truncation limitations.