CHAPTER V. 



Integration of the fundamental equation in statistical 



mechanics. 



32. Some theorems on the development of a frequency function in 

 a series of type A. 



We have use for some theorems from the mathematical statistics, which, for 

 the sake of completeness, are here deduced. 



Let us consider a series of type A of one variable: 



(66) F[x) = 2'/j„(p(")(ic) . 

 Putting 



(67) cf f")(^) = ^A^) ?W 

 and 



1 {x-LT 



so is (Meddel. N:o 27) 



(68) An = j ^(^) ■ 



— 00 



I will write in the following, until further notice, x instead of x — L. 



The functions Rn{x) are the polynômes of Hermite, their general expression being 



(CQ\ 7? f ^ — n{n — l) x"-^ n(n—l){n—2) {n—3) x'"^ _ 1 



(b.t) Unix) - |~ — 1^21 ^ 11^2 ~ ^ ~ •••} • 



Putting 



(— l^ 

 J?„(a:)=^i-^4x•), 



so that <S'„(aj) denotes the parenthesis in the expression (69), we have 

 (70) . F{x) = 'i{x) !to + Ti S,{x) + +...}, 



