795 research outputs found
A new approach to the epsilon expansion of generalized hypergeometric functions
Assumed that the parameters of a generalized hypergeometric function depend
linearly on a small variable , the successive derivatives of the
function with respect to that small variable are evaluated at
to obtain the coefficients of the -expansion of the function. The
procedure, quite naive, benefits from simple explicit expressions of the
derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols
with respect to their argument. The algorithm may be used algebraically,
irrespective of the values of the parameters. It reproduces the exact results
obtained by other authors in cases of especially simple parameters. Implemented
numerically, the procedure improves considerably the numerical expansions given
by other methods.Comment: Some formulae adde
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
We continue the study of the construction of analytical coefficients of the
epsilon-expansion of hypergeometric functions and their connection with Feynman
diagrams. In this paper, we show the following results:
Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth
(see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions.
Theorem B: The epsilon expansion of a hypergeometric function with one
half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the
harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are
ratios of polynomials. Some extra materials are available via the www at this
http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected
and a few references added; v3: few references added
Derivatives of Horn-type hypergeometric functions with respect to their parameters
We consider the derivatives of Horn hypergeometric functions of any number
variables with respect to their parameters. The derivative of the function in
variables is expressed as a Horn hypergeometric series of infinite
summations depending on the same variables and with the same region of
convergence as for original Horn function. The derivatives of Appell functions,
generalized hypergeometric functions, confluent and non-confluent Lauricella
series and generalized Lauricella series are explicitly presented. Applications
to the calculation of Feynman diagrams are discussed, especially the series
expansion in within dimensional regularization. Connections with
other classes of special functions are discussed as well.Comment: 27 page
Irregular Input Data in Convergence Acceleration and Summation Processes: General Considerations and Some Special Gaussian Hypergeometric Series as Model Problems
Sequence transformations accomplish an acceleration of convergence or a
summation in the case of divergence by detecting and utilizing regularities of
the elements of the sequence to be transformed. For sufficiently large indices,
certain asymptotic regularities normally do exist, but the leading elements of
a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1
(a, b; c; z) is well suited to illuminate problems of that kind. Sequence
transformations perform quite well for most parameters and arguments. If,
however, the third parameter of a nonterminating hypergeometric series 2F1
is a negative real number, the terms initially grow in magnitude like the terms
of a mildly divergent series. The use of the leading terms of such a series as
input data leads to unreliable and even completely nonsensical results. In
contrast, sequence transformations produce good results if the leading
irregular terms are excluded from the transformation process. Similar problems
occur also in perturbation expansions. For example, summation results for the
infinite coupling limit k_3 of the sextic anharmonic oscillator can be improved
considerably by excluding the leading terms from the transformation process.
Finally, numerous new recurrence formulas for the 2F1 (a, b; c; z) are derived.Comment: 25 pages, 7 tables, REVTe
Counting master integrals: Integration by parts vs. functional equations
We illustrate the usefulness of functional equations in establishing
relationships between master integrals under the integration-by-parts reduction
procedure by considering a certain two-loop propagator-type diagram as an
example.Comment: 8 pages, 1 figur
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