The current Index can be seen as offset by 1 due to starting at 1. If We look at K=1 and did not subtract 1 from the current index we would actually get 1+3(1) = 4 or 1+3(2)=7. This number increments each time across the "loop" and can be seen as similar to the Sigma∑ notation's looping functionality in that respect.) PreNote: ( k=1 is an index location, like finding a book in a library. The appearing of n or m as summation stop index implies n, m ∈ N.This is essentially a "hack" to avoid counting your current "index" location against the math. Shown above are the for n from 0 to 10 together with the results of corresponding Mathematica sums employing ' Abel' regularization as well as the symbolic HurwitzZeta given at the start of the paragraph. Isolating the term in (**) with the highest exponent (set the stop index in the sum over j in the table to n - 1) now allows a recursive calculation of the like:
#Find infinite sum of recursive sequence series#
The product inside the sum may be decomposed into a double series of StirlingS1 numbers : It determines at m = 3:Įxpanding the sum over (k+2) shows and, and if it is assumed that = 1/2 (regularization), then it follows that = - 1/4. This identity can be used to successively get values of.
Which gives after multiplication with (m - 1)! and evaluation
Then the lattice sum can be reduced to a single sum like This tells the number of ways to express k as a sum of m integers (how often a certain term ' k' occurs) ,įor instance (k = 3, m = 2 4 ways) : 3 = 0 + 3 or 3 + 0 or 1 + 2 or 2 + 1. Where the summands characterized by = k occur with a certain multiplicity given by Many of the series found in this table are connected to a 'lattice version' like (try it!) :įor instance (find an identity with start index 0, replace k →, insert the Gammas, the j - 1 factorial and sum over all i's) :įor a series starting with index 1 it is a little more involved, because the first lattice summand will be :įollowing ideas of (R4) look at the m-dimensional lattice sum Now the multiplicity of numerically equal summands ) is determined by Binomial:įor the lowest values of s and m this sum is:įor the lowest values of s and m this sum is : ] // FullSimplify // PowerExpand // Expand // TableFormįor a few values of s = 1 to 7 (rows) and m = 1 to 5 (columns) the results for the series above are given using a summation, that avoids ‘indeterminate’ answers. some properties of ProductLog LerchPhi and PolyLog Series involving Hypergeometric Functions Series of Zeta PolyGamma PolyLog and related Series of Inverse Tangents ( Arcustangent ) Sums involving reciprocal multifactorials or factorials special values of EllipticK and EllipticE Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series These pages list thousands of expressions like products, sums, relations and limits shown in the following sections: The site has been growing ever since, and its focus has been expanded After I learned that the double product can be solved usingĮlliptic theta functions I was hooked. Line charge trapped inside a rectangular tube. My interest in infinite products has its origin in the year 2000 in connection with the problem of the electrical field of a
Andreas Dieckmann, Physikalisches Institut der Uni Bonn Table of Infinite Products Infinite Sums Infinite Series Elliptic ThetaĬollection of Infinite Products and Seriesĭr.
0 kommentar(er)
