EXPANSION FOR FUNCTIONS OF HIGH ORDER 357 



mite with the Laplace polynomials have been presented in an earlier 

 paper. * 



Appell and Feriet, Arne Fisher, T. C. Fry, H. L. Rietz and others 

 base their definitions of the Hermite polynomials on e"^^/^ instead of 

 6"=^^. We shall write A„(u) for the «th polynomial as defined by these 

 authors, reserving Ilniii) to symbolize the Hermitian polynomial as 

 defined in his paper of 1864. Thus, in what follows, 



Au{x) = (- l)"e-/2(<ine-x2/2/^^„)^ /^^(^) = (_ iV2)M„(MAr 



III 



Laplacian expansions * for the U, H, and A polynomials follow 

 immediately from those obtainable by applying to the integrals /„(«) 

 and In{u) his method of evaluating definite integrals whose integrands 

 embrace factors raised to high powers. As will be shown in Part IV 

 of this paper, we have 



In{u)|l-fK{Y^[Ny] = [5 cos (mV27V)] + [5' sin (uyjlNJi, N = 2n, 

 V(w)/[V7r(FViV)^] = [5 sin {u\'2N)2 - IS' cos {u-ilN'}, N =2n + \, 

 where Y = (.rg-^^) for x = X = Xj^ll and 



The explicit expressions for Kq, Ko, Ki and K\, K3, K^ are given in 

 Section V of this paper. The desired expansions are then given by 

 the equations: 



e-/„(M)/V^ - Z7„(«)[(2«)!/22»+i«!] 



= 7/2.(«)[(- l)"/2'"+^] 



= ^2n(wV2)[(- l)"/2"+a 



e"V„'(«)/V^= Un'{u) liln + l)!/22"+i«!] 



= ^2«+i(wV2)[(- l)»/2"+'>/2]. 



The numerical results shown below in Table I indicate the efficacy 



* It may be of interest to compare the expansions presented in this paper with 

 the asymptotic forms of the Hermite functions given by N. Schwid* and by M. 

 Plancherel and M. Rotach.'" 



