I would like to apply Ramanujan's Master Theorem (RMT) to formally justify some integrals that I have been using. The source for the proof of the RMT that I have is Hardy's book on Ramanujan's work. The formal statement of the RMT can be found in https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.704.4327&rep=rep1&type=pdf as Theorem 3.2. In Hardy's book, he says that the growth condition on $\varphi$ (and in particular the requirement $A < \pi$) is 'natural', but insufficient in many practical applications, where apparently $A=\pi$ is the best bound available.
In almost all practical applications I have seen, the formal statement of the RMT is more or less ignored and it is applied 'blindly' to derive a bunch of interesting integrals.
How can one generically check that the growth condition on $\varphi$ is satisfied? Is there a simpler condition which implies this kind of growth?
Specifically, I am interested in proving entry 3.252-10 in Gradsteyn & Rhyzik: http://fisica.ciens.ucv.ve/~svincenz/TISPISGIMR.pdf
If I expand the function $$f(x) = \frac{1}{(1+2x\cos t +x^2)^{\alpha_2}}; \quad (\alpha_2 >0) $$ in terms of Gegenbauer polynomials and 'blindly' apply the RMT, I get the correct answer (after some trivial manipulations and using the generalized definition of Gegenbauer polynomials in terms of ${}_2F_1$, which can also be found in the above link).
To meet the conditions of the RMT, I would need to prove that the function $$ \phi(s) = C^{\alpha_2}_{s}(\cos t) $$ satisfies said growth condition as a function of $s$, where $0 < t < \pi$. I don't know how to approach this.
Edit: for completeness' sake, the growth condition reads: for some $0<\delta<1$ and $s=v+iw$ in the half-plane $v \geq -\delta$ we have $$|\phi(v+iw)| \leq C \exp(Pv + A|w|)$$
for some $P$ and $A<\pi$.
Any help is appreciated!