Plana’s summation formula for $\sum\sp \infty\sb {m=1,3,\cdots}m\sp {-2}\sin(m\alpha),m\sp {-3}\cos(m\alpha),m\sp {-2}A\sp m,m\sp {-3}A\sp m$
نویسندگان
چکیده
منابع مشابه
Euler Summation Formula for Path Integrals
We present and comment on some details of a new analytical method for systematic improvement of the convergence of path integrals of a generic N -fold discretized theory. The new methods represents a Euler summation formula for path integrals. Keeping the first p terms in this formula improves convergence of path integrals to the continuum limit to 1/N. We have given explicit calculations up to...
متن کاملA Multiparameter Summation Formula for Theta Functions
We generalize Warnaar's elliptic extension of a Macdonald multi-parameter summation formula to Riemann surfaces of arbitrary genus. We start by a brief outline of the general theory of hypergeometric type series built from Jacobi theta functions [12, 13] (its extension to integrals [15] is not touched at all). Within this approach univariate elliptic hypergeometric series are defined as the ser...
متن کاملCurious Extensions of Ramanujan’s 1ψ1 Summation Formula
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our identities, by analytic continuation, to bilateral summation formulae which contain Ramanujan’s 1ψ1 summation and a very-well-poised 4ψ6 summation as special c...
متن کاملA New Multivariable 6ψ6 Summation Formula
Abstract. By multidimensional matrix inversion, combined with an Ar extension of Jackson’s 8φ7 summation formula by Milne, a new multivariable 8φ7 summation is derived. By a polynomial argument this 8φ7 summation is transformed to another multivariable 8φ7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6φ5 summation. The latter is ...
متن کاملResurgence of the Euler-MacLaurin summation formula
Abstract. The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Pla...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1990
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1990-1035929-9