192 



Mathematics. — "On Hermite's and Abel's polynomia.'" By 

 N. G. W. H. Beeger. (Communicated by Prof. W. Kapteyn). 



(Communicated in the meeting of May 30, 1914). 



Prof. Kapteyn has deduced the following expansion ^) : 



— : :r- 6 =^ ..... (11 



in which Hn [x] represent the polynomia of Hermite. Let in this 

 expansion « = 0, then we find : 



e 



1-62 _^ 6nHn{x)Hn{0) .... (2) 



1/(1— <9^) ' ~~ 2» . n .' 



Now it holds good for the polynomia of Hermite that: 



^2«+i(0) = iy2„ (0) z= (— 1)« ^V^ ... (3) 



n! 



On account of which the above relation passes into: 



62x2 



e 



= È{-\Y^^6^n (4) 



\/{l-0') 22». n.' 



For the polynomia cpn G?') of Abel we know the expansion : 



a.-«6 



1 — tf 



If we replace in (4) 6^ by 6 we find: 



9a,-2 



e =2( — IV» ^ &n 



^/(l— <9) 22". n.' 



If we multiply the first member of this relation by and 



^ [/{\—0) 



X (2n).' 



the second member by -^ ó»", the first member becomes equal 



22'». (n/)» 1 



to the first member of (5). ^y equalizing the coefficient of 6" in the 



two second members, we find the following relation between the 



polynomia of Abel and those of Hermite: 



If we multiply both members of (6) by 



H^ ,1 _ i (.r ) e—^'^ dx 



1) These Proceedings. Vol. XVI, p. 1198 (22). 



