Integral Representation and Algorithms for Closed Form Summation

Wilf′s Conjecture

“…It follows that P 6[4] − I is strictly lower triangular (mod 2). Repeating the same argument, B = P 12[8] − I is strictly lower triangular. Hence from Lemma 8, P 48[8] − I ≡ B 4 (mod 2) is strictly lower triangular (mod 2), and its first three lower diagonals are even.…”

“…(b) All upper diagonals, the main diagonal, and the three lower diagonals of P d 4 [8] − I are even.…”

“…Klazar asked the same question in the MONTHLY note [11] mentioned before. In a different context, this question is relevant to a result of Egorychev and Zima [8], who showed that the sum n k=1 k a k! admits a closed form expression in terms of elementary functions only if B ± (a + 1) = 0.…”

New applications of the Egorychev method of coefficients of integral representation and calculation of combinatorial sums

“…At the end of the 1970's G.P. Egorychev developed the method of coefficients, which was successfully applied to many combinatorial problems [11,13,9,14] and [42]. Here we present the (see section 2.1) short description of the Egorychev method and its recent applications to several problems of enumeration and summation in various fields of mathematics: algebra, the theory of integral representations in C n and the theory of approximation.…”

Integral Representation and Computation a Multiple Sum in the Theory of Cubature Formulas

“…At the end of the 1970's, G.P. Egorychev has developed the method of coefficients, which was successfully applied to many combinatorial sums [1,2,3,6]. The purpose of this article is finding a new simple proof of identity (1.1) by means of the method of coefficients [1] and multiple applications of a known theorem on the total sum of residues in the theory of holomorphic functions.…”