Wikipedia:Reference desk/Archives/Mathematics/2009 March 13

= March 13 =

Products of Multiple Geometric Series
Does anybody know a way to simplify

: | |     \       :  | |     /      (Ri)-j : | |    -- :  i=1      j=0                               ?

Sorry about the ASCII approximations of symbols, but I don't speak LaTex... Lucas Brown 42 (talk) 19:21, 13 March 2009 (UTC)
 * Set ri = 1/Ri. For finite m, I think you are stuck with just foiling it out along the lines of the multinomial theorem.  For infinite m you get a formal equality:
 * $$\prod_{i=1}^n \sum_{j=0}^{\infty} r_i^j = \sum_{j=0}^{\infty} e_j(r_1,r_2,\dots,r_n)$$
 * where ej is the elementary symmetric polynomial of degree j. JackSchmidt (talk) 19:39, 13 March 2009 (UTC)


 * That can't be right since for $$e_j(r_1,r_2,\dots,r_n)$$ we must have $$j \leqslant n$$. Dauto (talk) 18:42, 14 March 2009 (UTC)


 * Would you believe: For infinite m you get a formal equality:
 * $$\prod_{i=1}^n \sum_{j=0}^{\infty} r_i^j = \sum_{j=0}^{\infty} h_j(r_1,r_2,\dots,r_n)$$
 * where hj is the complete homogeneous symmetric polynomial of degree j. Whichever one is the one with each variable appearing to each power. JackSchmidt (talk) 03:07, 15 March 2009 (UTC)