( 749 ) 
Mathematics. — “An integral-theorem of Gwaenpaunr.” By Prof. 
KLUYVER. ; 
(Communicated in the meeting of February 27, 1909). 
GEGENBAUER has proved a theorem according to which the product 
of two functions of Brssrr /’*(ax) and /’(bx) with the same para- 
meter v >— 4 can be given the form of a definite integral '). 
In a former communication *) I have applied this theorem for the 
case r =Q(0 when reducing some discontinuous integrals containing 
functions of Brssen. [ shall now give in the following a direct proof 
of the indicated theorem and shall use it to extend former results. 
1. In order to find the product of two functions of Besser 
aN On 
ea 
7) 2 
Poa) = (5 DME 
(—1) (=) 
el me 1)? 
ba \’ q 
we can multiply the absolutely converging power series. It is then 
evident, that (supposing 6< a) we find for the coefficient of an 
arbitrary power of x a finite hypergeometic series with the fourth 
b: 
argument —. 
The 
We get’) 
a \2h 
109 lee Ty a es 
aba Na 2 b? 
J (ax) J’ (bx) =| —— N ach | —vy-h,-h,y +1, ;) 
C 
4 am HI (v4 LT HAI) 1 
To transform the hypergeometric series appearing here I shall use 
the notation of Riemann for the general hypergeometric function 
a kt 
Ea @ ÔrYs A k 
Paed >| 
Et PB + y tr ==) 
Pte er [le A. (2 —a) J- Alea) Arsis |. 
1) Nietsen. Handbuch der Theorie der Cylinderfunktionen, page 182. 
2) Proceedings 1905. 
5) Nietsen. page 20. 
51 
Proceedings Royal Acad. Amsterdam. Vol. XI. 
