# Euler/Pfaff transformations for generalized hypergeometric functions $_pF_{p+1}$ functions

For hypergeometric function $_2F_1(a_1,a_2;b_1;z)$ there exists Euler/Pfaff transformations: $$_2F_1(a_1,a_2;b_1;z)=((1-z)^{b_1-a_1-a_2})_2F_1(b_1-a_1,b_1-a_2;b_1;z),\quad \text{Euler transformation}$$

$$_2F_1(a_1,a_2;b_1;z)=((1-z)^{-a_1})_2F_1\left(a_1,b_1-a_2;b_1;\frac{z}{z-1}\right),\quad \text{Pfaff transformation}$$

$$_2F_1(a_1,a_2;b_1;z)=((1-z)^{-a_2})_2F_1\left(b_1-a_1,a_2;b_1;\frac{z}{z-1}\right),\quad \text{Pfaff transformation}$$

We are looking forsimilar formulas for $_1F_2(a_1;b_1,b_2;z)$, $_2F_3(a_1,a_2;b_1,b_2,b_3;z)$, and $_3F_4(a_1,a_2,a_3;b_1,b_2,b_3,b_4;z)$.

Any comments and references are welcome!

Thanks- mike

• Sort of the opposite: "Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x)" and (which also misses by an operation) "SERIES TRANSFORMATION FORMULAS OF EULER TYPE, HADAMARD PRODUCT OF FUNCTIONS, AND HARMONIC NUMBER IDENTITIES" I think I remember also using the standard integral representations in a non-standard way (Beta integrals) to do something similar if you are still interested I will try to recall it. – rrogers Aug 20 '16 at 17:37
• @rroges: thanks a lot for your comment! Currently I do not need these perticular expressions anymore. Best – mike Aug 21 '16 at 0:53
• Actually I think I have found a way to generally find similar relations but it's to long (and a little shaky) to put in the margin:) – rrogers Apr 22 '17 at 12:44