What are the key properties of the Generalized Beta Function and its integrals?

[alyafey22]

In this thread we consider the integrals of the form

$$\beta(a,b,c;s) = \int^1_0 \frac{1}{(1-x)^a(s-x)^b x^c}\, dx \,\,\,\,\,\,-1<a,b,c<1 \,\,\,s\geq 0 $$

This is NOT a tutorial , all suggestions are encouraged.

 

[alyafey22]

We can relate it to the beta function $$s=1$$
Assume $$\Re(a+b)>0 , \Re(1-c)>0$$

$$\beta(a,b,c;\,1) = \int^1_0 (1-x)^{-(a+b)} x^{-c}dx= \beta(1-c,1-(a+b))=\frac{\Gamma(1-c)\Gamma(1-(a+b))}{\Gamma(2-(a+b+c))}$$

$$\tag{1} \beta(a,b,c;\,1) = \frac{\Gamma(1-c)\Gamma(1-(a+b))}{\Gamma(2-(a+b+c))} \,\,\,\, \Re(a+b)>0 , \Re(1-c)>0$$​

 

[DreamWeaver]

In this thread we consider the integrals of the form

$$\beta(a,b,c;s) = \int^1_0 \frac{1}{(1-x)^a(s-x)^b x^c}\, dx \,\,\,\,\,\,-1<a,b,c<1 \,\,\,s\geq 0 $$

This is NOT a tutorial , all suggestions are encouraged.

You're on tip-top form, Zaid! I like it! (Heidy)

A few things jump out at me, which might be worth exploring...

Firstly, the reflection substitution $$x \to 1-x$$ might be worth a look... Also, being the logarithmic fiend that I am, I think it might be worth considering partial derivatives wrt any/all of the parameters.

I'll definitely come back to this topic when it's not so close to bed time. Very interesting! (heart)