Inequalities and the growth of hypergeometric functions with respect to the coefficients

Question

For a question in geometry I was led to the following questions on hypergeometric functions. For $a>0$, let $$h_a(x)={}_2F_1(a i, -a i; 1; x)$$ where $i$ is a root of $-1$. Here $h_a(1)=\frac{\sinh a}{a}$ is well-defined. Consider $f(x)=x(1-x)h_a'(x)$ which is $=0$ at $x=0, 1$. Is it true that there exists a uniform constant $C$ such that for all $a>0$, $$f(x)=\frac{x(1-x)h_a'(x)}{h_a(x)} <C$$ for all $x \in [0,1]$? Numerically, I have used SageMath for $a \leq 500$ and this ratio seems to be quite small. If not true, it is possible to derive the growth rate of the ratio with respect to the coefficient $a$?

I have tried many formulas for hypergeometric functions (Contiguous Relations) in this book Special Functions without success. I also have tried just plain Taylor expansion and it is difficult to see any bounds. Thank you very much!

Answer 1

There is something that I do not understand. $$f_a(x)=a^2x(1-x)\frac{\, _2F_1(1-i a,1+i a;2;x)}{\, _2F_1(-i a,i a;1;x)}$$ For large values of $a$, the maximum of the function is attained closer and closer to $x=\frac 12$.

Using this asymptotic value $$f_a\left(\frac{1}{2}\right) = \frac a 2 -\frac 14+O\left(\frac{1}{a}\right)$$ which seems also to be true for $f_a\left(x_{\text{max}}\right)$

For example, $$a=2 \implies x_{\text{max}}=0.530676 \implies f_2\left(x_{\text{max}}\right)=0.767445$$ $$a=3 \implies x_{\text{max}}=0.513503 \implies f_3\left(x_{\text{max}}\right)=1.264525$$

Answer 2

I have tried three definitions, each yielding some other nonsense. The imaginary $i\, a$ is the exponent in the definition of Jacobi (Vol 6)

$$\;_2F_1(\alpha,\beta,\gamma,x) = \int_0^1 \ \frac{z^\alpha\ (1-z)^\beta}{(1- x\ z)^\gamma} dz$$

with a purely imaginary

$$\;_2F_1(i\, a,-i\ a, 1,x) = \int_0^1 \ \left(\frac{z }{1-z}\right)^{i a} \frac{dz}{1- x z}$$

that is defined by its exponential, as we know today

$$\;_2F_1(i\, a,-i\ a, 1,x) = \int_0^1 \ e^{i\, a \log\left(\frac{z }{1-z}\right)} \frac{dz}{1- x z}$$

Mathematica yields

$\text{Assuming}\left[-1<\Im(a)<1\land 0<x<1,H_{\text{a$\_$}}(\text{x$\_$})=\int_0^1 \frac{\left(\frac{t}{1-t}\right)^{i a}}{1-t x} \, dt\right]$

$$\frac{i \pi \left(1-(1-x)^{-i a}\right) \text{csch}(\pi a)}{x}$$

This expression has nothing to do with Hypergeometric2F1[I a, -I a, 1, x]]

By a look at Gradshteyn/Rhyzik all integral formulas demand $\Re \alpha, \beta >0 $. On the other hand, the formal series for $\gamma=1$ absolutely converges in the unit circle.

An integral formula with an imaginary power generally needs special methods from functional analysis, especially the question how to implement numerical approximations for expressions nobody knows about.

Parametric integrals are difficult to implement as numeric functions, Plot etc don't accept them as non-numeric.