Binomial Theorem If n is a positive integer, then (a + x )n = a n + nC 1a n ?1x + nC 2a n ?2x 2 + nC 3a n ?3x 3 + where nC r = + xn n! . r!(n ? r )! If n is not a positive integer, then (1 + x )n = 1 + nx + where -1 < x < 1. n(n ? 1...
Rothe formulaeBy means of the classical Lagrange expansion theorem, five convolution formulae are established for the orthogonal polynomials named after Laguerre, Jacobi, Meixner, Gegenbauer and Pollaczek, that contain the well-known Hagen–Rothe formula for binomial coefficients as common special case...
Furthermore, using the q-exponential operator technique to the multiple q-binomial theorem a... ZM Liu - 《Discrete Mathematics》 被引量: 54发表: 2006年 Generalization of Kummer's second theorem with applications The results are derived with the help of generalized Gauss second summation theorem...
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Electrical formulae for current, voltage, resistance, power and conductivity.The units and formula for various electrical parameters included in the page. Solved Examples gives a better understanding of the concepts like current, voltage, resistance etc.
In this appendix, summation formulae for series involving nested sums and binomial coefficients are presented. Some of them are used in this book. These series have a canonical form, in the sense that the argument of a nested sum \\\(S_... VA Smirnov - Springer Berlin Heidelberg 被引量:...
certain properties)ofsuchasystem.Hewasthusabletoderiveanintegralrepresentationforthis 2 Φ 1 seriesandhencegiveanalternativeproofoftheq-Selbergintegral[2](seealso[9,23,10]). Headditionallyderivedtheq-analogueoftheGaussformulaandanotherintegralformula,the constanttermversionofwhichwaspresentedin[16,Theorem...
Math SL Formulae IB
The general result (see Theorem 4.1) holds for any group of the form ZN where N鈭圢 and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing N to different values, we see that these expressions simplify in some cases and we obtain ...
By establishing new multiple inverse series relations (with their connection coefficients being given by higher derivatives of fixed multivariate analytic functions), we illustrate a general framework to provide new proofs for MacMahon’s master theorem and the multivariate expansion formula due to Good ...