TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 20 



we may write Q(j) under the form, 



/ n— s , V' /+ V n^-l—s , V v+1 



(-T-+*). (--!-+») . 



With these expressions of A, QO), P(^i), we have now 



S(u )V ) = A[2^ 2' ""i]Q(i)P(ii); 

 or, as the limits of i, j are independent from one another, we may intervert so that 



where r is to be replaced by r' as the case may demand. 



§ HI 



We are now arrived at a stage where we have to consider generally the transforma- 

 tions of sums of the form 





l'/+y/-e" 



which we shall designate by 



We shall assume that the number of the terms be limited in consequence of the 



hypothesis 



a = integer negative. 



By this the upper limit t x of t will be 



t x = —a. 



Of course we must also assume that the factors in the denominator do not pass through 

 zero for any value of t between its limits. This involves the hypothesis that y, or e, or 

 both, should they be integer negative, must be in absolute value greater than — a. 



For the transformation of F our auxiliary will be a particular case of Gauss's 

 function,""' 



f(a, ft y, x) = |2" ijp+i^r+l*'- 

 Our particular case is of course 



05=1, 



a = integer negative. 



* Commentatio7ies Societatis Goettingensis recentiores, to. ii., anno 1812. 



