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

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March 13

Products of Multiple Geometric Series

Does anybody know a way to simplify

:   n        m
: -----   ------ 
:  | |     \      
:  | |     /      (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 r_i = 1/R_i. 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 e_j 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 h_j 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)