3,259 research outputs found
New Series Expansions of the Gauss Hypergeometric Function
The Gauss hypergeometric function can be computed by using
the power series in powers of . With
these expansions is not completely computable for all
complex values of . As pointed out in Gil, {\it et al.} [2007, \S2.3], the
points are always excluded from the domains of convergence
of these expansions. B\"uhring [1987] has given a power series expansion that
allows computation at and near these points. But, when is an integer, the
coefficients of that expansion become indeterminate and its computation
requires a nontrivial limiting process. Moreover, the convergence becomes
slower and slower in that case. In this paper we obtain new expansions of the
Gauss hypergeometric function in terms of rational functions of for which
the points are well inside their domains of convergence . In
addition, these expansion are well defined when is an integer and no
limits are needed in that case. Numerical computations show that these
expansions converge faster than B\"uhring's expansion for in the
neighborhood of the points , especially when is close to
an integer number.Comment: 18 pages, 6 figures, 4 tables. In Advances in Computational
Mathematics, 2012 Second version with corrected typos in equations (18) and
(19
Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Poschl-Teller-Ginocchio potential wave functions
The fast computation of the Gauss hypergeometric function 2F1 with all its
parameters complex is a difficult task. Although the 2F1 function verifies
numerous analytical properties involving power series expansions whose
implementation is apparently immediate, their use is thwarted by instabilities
induced by cancellations between very large terms. Furthermore, small areas of
the complex plane are inaccessible using only 2F1 power series formulas, thus
rendering 2F1 evaluations impossible on a purely analytical basis. In order to
solve these problems, a generalization of R.C. Forrey's transformation theory
has been developed. The latter has been successful in treating the 2F1 function
with real parameters. As in real case transformation theory, the large
canceling terms occurring in 2F1 analytical formulas are rigorously dealt with,
but by way of a new method, directly applicable to the complex plane. Taylor
series expansions are employed to enter complex areas outside the domain of
validity of power series analytical formulas. The proposed algorithm, however,
becomes unstable in general when |a|,|b|,|c| are moderate or large. As a
physical application, the calculation of the wave functions of the analytical
Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.Comment: 29 pages; accepted in Computer Physics Communication
Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
We examine the expansions of the solutions of the general Heun equation in
terms of the Gauss hypergeometric functions. We present several expansions
using functions, the forms of which differ from those applied before. In
general, the coefficients of the expansions obey three-term recurrence
relations. However, there exist certain choices of the parameters for which the
recurrence relations become two-term. The coefficients of the expansions are
then explicitly expressed in terms of the gamma functions. Discussing the
termination of the presented series, we show that the finite-sum solutions of
the general Heun equation in terms of generally irreducible hypergeometric
functions have a representation through a single generalized hypergeometric
function. Consequently, the power-series expansion of the Heun function for any
such case is governed by a two-term recurrence relation
Large Parameter Cases of the Gauss Hypergeometric Function
We consider the asymptotic behaviour of the Gauss hypergeometric function
when several of the parameters a, b, c are large. We indicate which cases are
of interest for orthogonal polynomials (Jacobi, but also Krawtchouk, Meixner,
etc.), which results are already available and which cases need more attention.
We also consider a few examples of 3F2-functions of unit argument, to explain
which difficulties arise in these cases, when standard integrals or
differential equations are not available.Comment: 21 pages, 4 figure
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