All Questions
16 questions
2
votes
1
answer
188
views
Incomplete integral of confluent hypergeometric function
I would like to prove the following: if $a,b,c > 0$, $0<x<1$, $y>0$, then,
$$
\frac{1}{\Gamma(b)}\int_0^{\infty} s^{b-1} e^{-s} \,\mbox{$ {}_1\mathrm{F}_1 (a\,; c\,; x s + y\hspace{1pt})$}\...
- 391
1
vote
1
answer
197
views
Derivation of indefinite integral involving hypergeometric function
I am doing a project on projectile motion and I ended up with this integral:
$$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$
where $g, c,$ and $m$...
- 21
1
vote
0
answers
197
views
Infinite series involving generalised hypergeometric functions
I've recently stumbled into hypergeometric functions while trying to evaluate the integral:
$$
\int \exp \big( x^2 + bx + c \big) {\rm erf} ( x ) \operatorname{d\!}x
$$
Essentially, working from an ...
- 41
0
votes
1
answer
128
views
Generalization of identity for terminating hypergeometric function
Let ${}_2F_1(a,b;c;z)$ be the ordinary hypergeometric function for $z \in \mathbb{C}$
\begin{equation}
{}_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{z^k}{k!} \frac{(a)_{k} (b)_k}{(c)_k}\,,
\end{...
- 679
2
votes
0
answers
81
views
Uniform bound on a certain family of hypergeometric functions
We have the following problem, which we can't solve.
Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
- 5,562
1
vote
1
answer
173
views
Higher-order asymptotics of generalized hypergeometric function
I have a question about higher-order asymptotics of generalized hypergeometric functions. According to https://dlmf.nist.gov/15.4
the following is well known:
$$
_2F_1(a,b;a+b;z)\sim -\frac{\Gamma(a+b)...
11
votes
1
answer
566
views
Integral representation of product of two Whittaker functions
Does anyone know anything about the following formula involving special functions:
$$\begin{multline*}
W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
- 373
3
votes
2
answers
266
views
Integral expressions for Bessel-like power series
I'm interested in power series of form $$f(z)=\sum_{k=0}^\infty \frac{z^k}{(k!)^\alpha}.$$ When $\alpha=1$, this becomes $\exp(z)$. For $\alpha=2$ this is a Bessel function and for larger integer $\...
- 1,324
2
votes
1
answer
478
views
Correction terms in the asymptotic expansion of hypergeometric function
I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below),
$$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\frac{1}...
- 21
3
votes
1
answer
599
views
Compute Confluent Hypergeometric Function 1F1
I am attempting to compute the (Kummer's) confluent hypergeometric function (see also here)
\begin{align}
M\left(\frac{n}{2}, n +\frac{3}{2}, -z\right) = {}_1F_1\left(\frac{n}{2};n +\frac{3}{2};-z\...
- 61
3
votes
0
answers
106
views
Does the Riemann characterization of the hypergeometric function have a q-analog?
This question is inspired by another recent one here, Characterization of the hypergeometric function. The latter is about the classical result of Riemann characterizing the hypergeometric functions ...
- 18.4k
4
votes
1
answer
211
views
Perform an integration over the unit interval of a two-parameter expression involving a Gauss hypergeometric function
In a quantum-information-theoretic context, I've encountered the problem
of integrating over $r \in [0,1]$, the function
\begin{equation}
r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2}...
- 723
6
votes
1
answer
363
views
Double Series involving Gamma function
Does anyone have any ideas on howto verify $$\sum_{n,m=0}^\infty \frac{\Gamma(n+m+3x)}{\Gamma(n+1+x)\Gamma(m+1+x)}\cdot \frac{1}{3^{n+m+3x-1}} = \Gamma(x)$$ for $x>0$?
I posted this question also ...
- 284
5
votes
0
answers
388
views
Is a basic hypergeometric function ${}_2\phi_1(a, b; c; q, z)$ a meromorphic function in $z$?
Here a basic hypergeometric function is the analytic continuation of the basic hypergeometric series (or called the $q$-hypergeometric series)
$$
{}_2\phi_1(a, b; c; q, z) = \sum^{\infty}_{n = 0} \...
- 123
4
votes
2
answers
522
views
Analytic continuation of $_4F_3(1)$
The Gauss theorem
$${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$
allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series ...
- 41
40
votes
1
answer
2k
views
Curious $q$-analogues
Consider the Fibonacci polynomials
$$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$
and the Lucas polynomials
$$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \...
- 5,622