356 BELL SYSTEM TECHNICAL JOURNAL 



These polynomial functions formed the coefficients in a series sat- 

 isfying the partial differential equation 



dr du du^ 



to which Laplace reduced the solution of the following ball problem: ' 



Consider two urns A and B each containing n balls and suppose that of the total 

 number of balls, 2m, as many are white as black. Conceive that we draw simultane- 

 ously a ball from each of the urns, and that then we place in each urn the ball drawn 

 from the other. Suppose that we repeat this operation any number, r, of times, 

 each time shaking well the urns in order that the balls be thoroughly mixed: and 

 let us find the probability that after the r operations the number of white balls in 

 urn A be x. 



Under the caption "The Statistical Meaning of Irreversibility" 

 Lotka * has pointed out the significance of Laplace's ball problem in 

 the modern kinetic theory of matter. Moreover, Hostinsky ^ has 

 shown the bearing of the same problem on the theory of Brownian 

 movements and said "In effect, the partial differential equation ob- 

 tained by Laplace has been refound by Smoluchowski." 



To avoid confusion with the Laplace functions which one encounters 

 in spherical harmonic analysis, the functions defined by Equations (1) 

 and (2) are herein designated as Hermitian-Laplace functions. Such 

 a designation is justified by the Equations (3) and (4) derived in the 

 next paragraph. 



II 



We also find in Laplace ^ 



In{u)* — j e^'^x^"- COS {2tix)dx = ' 







22"+i \ du^" 



L:{u) = f e-^V"+^ sin (2ux)dx = ^ ^^ ^ V^i • 



Comparing these Laplacian expressions for the definite integrals I„{u) 

 and /„'(w) with the Equations (1) and (2) we see immediately that 



(3) Un{u) = (- l)"[«!/(2«)!]iJ2n(«), 



(4) t//(«) = (- l)"+i[w!/2(2» + l)!]i/2„4-iW, 



where Hzn and H2n+i are the original Hermite polynomials^ of order 

 2« and 2n + 1, respectively. These equations connecting the Her- 



* The symbols In{n) and In'{u) are introduced here as convenient abbreviations 

 for the integrals to which they are equated; these symbols do not appear in Laplace. 



