Questions tagged [hypergeometric-functions]
Hypergeometric functions are the analytic functions defined by Taylor expansions of the shape $\sum_{n \geq 0} a_n x^n$, where $a_{n+1}/a_n$ is a rational function of $n$. This general family of functions encompasses many classical functions. The hypergeometric functions play an important role in many parts of mathematics.
268
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Antiderivative of $f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$
I need to know the primitive function (Antiderivative) of this function:
$$f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$$
where
$K$ is an integration constant,
$d=-\frac{1}{2p}$ with $p<...
3
votes
1
answer
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Generating function of the product of Legendre polynomials
The generating function of the product of Legendre polynomials for the same $n$ is given by
\begin{aligned}
\sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
3
votes
0
answers
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Functional inverse of $z=1+w+\cdots+w^{n-1}$
Migrated from the MSE.
I am interested in the functional inverse of
$$
z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1.
$$
This function is strictly increasing on $w\geq0$ and thus admits an inverse.
By ...
1
vote
1
answer
131
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Estimates for the absolute value of the hypergeometric function ${}_2F_1(2-n,n+2,2;x)$ on $[0,1]$
I would like to know whether the estimate
$$
|{}_2F_1(2-n,n+2,2;x)| \le \frac{n}{2}, \quad x \in [0,1]
$$
holds and in that case where to find a reference.
1
vote
1
answer
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Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function
Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series
$$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
7
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On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series
The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series:
\begin{align*}
\int_{-\infty}^{\...
1
vote
0
answers
60
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Integrability of an alternating series with hypergeometric coefficients
during my research, I came up with the following series
$$
f(t):=\sum_{k=0}^\infty \frac{\left(-t^2\right)^k}{(k!)^2}{}_3F_{2}\left(\left(-k,-k,-k\right);\left(1,\frac{1}{2}-k\right);\frac{1}{4}\right)...
3
votes
2
answers
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$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function
I try to calculate the following series
\begin{align*}
S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k},
\end{...
0
votes
1
answer
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Hyper geometric series reference
Can someone point out a reference for the proof of this identity? Thanks in advance.
https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric4F3/03/03/01/0002/
5
votes
0
answers
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Is there a special function for $\sum_{m=0}^{\infty} x^m/(m!)^s$?
Is there a special function for the following series?
$$\sum_{m=0}^{\infty} {x^m \over (m!)^s}$$
Here, $s$ is a positive real number.
When, $s$ is an integer, $s=n \in \mathbb{Z}$, this series can be ...
4
votes
1
answer
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Proof identity for hypergeometric series 2F1(a,b;c;x)
I would like prove the following identity:
$$
_2F_1(a,b;c,x) = \frac{c+(a−b+1)x}{c} {} _2F_1(a+1,b;c+1;x) - \frac{(a+1)(c−b+1)x}{c(c+1)} {} _2F_1(a+2,b;c+2;x).
$$
I've tried this so far:
I know that $...
0
votes
1
answer
102
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How to show the following inequality $_2F_1(5.5, 1, 5;-|x|^2]>0$? [closed]
Consider the function $_2F_1(5.5, 1, 5;-|x|^2]$ for $x\in \mathbb{R}^n.$ I want to show that this function is positive. I checked that it does not have any roots so can I conclude the inequality by ...
1
vote
0
answers
116
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Identities on the Whittaker function $W_{-\kappa,\mu}(z)$?
As in (for example) [Magnus, W., Oberhettinger, F., Soni, R.: Formulas and Theorems for the Special Functions of Mathematical Physics, Grundlehren der mathematischen Wissenschaften 52, Springer, ...
2
votes
0
answers
51
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Reduction of the general Lauricella hypergeometric function $F_B$ for identical parameters and variables
The Lauricella function $F_B^{n}$ of $n$ variables is defined as $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...
3
votes
1
answer
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6-j symbols and hypergeometric series
What’s the correct formula for $_{4}F_{3}(a,b,c,d;e,f,g;1)$ where $a+b+c+d-e-f-g=-1$?
The Wolfram Alpha formula involves $6j$ symbols and makes no sense for some specific cases. For example, $_{4}F_{...
1
vote
0
answers
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Any known relations to this doubly exponential constant?
Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1:
$$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \...
1
vote
0
answers
170
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Asymptotics of $\sum \frac{d(n)}{n}$ with generating functions
We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\...
5
votes
1
answer
511
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Hypergeometric function evaluation 4F3
I need to show that for $m$ being non-negative integer,
the hypergeometric function ${}_4F_3$ below evaluates to $-1/2$ independent of $m$.
This is Mathematica notation, but we have 4 and 3 sets of ...
1
vote
0
answers
98
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Asymptotic behavior of hypergeometric function ${}_{3}F_{2}(a,b-n,c-n;d-n,e-n;1)$ for $n\to\infty$
Suppose $a,b,c,d,e\in\mathbb{R}$ are such that $d+e-a-b-c>0$ and $d,e\notin\mathbb{Z}$. I would like to know whether it is possible to deduce an asymptotic formula for the sequence given by the ...
3
votes
1
answer
261
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Periodic Gauss hypergeometric function
I stumbled upon the following identity, which I have not tried to prove, but seems true:
the function $$f(t):={}_2F_1(1/2,2t;1-t;4)$$
is periodic of period 1, and more precisely
$$f(t)=\dfrac{1+2e^{-2\...
9
votes
0
answers
420
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What is the precise definition of "Hypergeometric motives over $\mathbb{Q}$"?
The question is as in the title, but here is some background:
Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
5
votes
1
answer
275
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Relationship between Lambert $W$ function and Hypergeometric function
The Lambert $W$ Function is defined in this Wikipedia entry, while the Hypergeometric Function is defined in this other Wikipedia entry. There exists also a multivariate generalization which solves ...
2
votes
0
answers
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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}_{...
4
votes
2
answers
428
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About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions
NOTE: I post this question on math.stackexchange but nobody answered, so I try here.
For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
47
votes
6
answers
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Are hypergeometric series not taught often at universities nowadays, and if so, why?
Recently, I've become more and more interested in hypergeometric series. One of the things that struck me was how it provides a unified framework for many simpler functions. For instance, we have
$$ \...
6
votes
0
answers
146
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Distribution of iid hypergeometric random variables conditioned on the sum
Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific,
$$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$
Let $S=X_1+\cdots+X_n$....
-2
votes
1
answer
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Hypergeometric function with changed argument [closed]
I have the hypergeometric function $_2F_1 (a, b,c, p\cdot z)$, where $p$ is a parameter and $z$ is the independent variable. I would like to know how the former function is related to the standard ...
9
votes
0
answers
318
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When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
0
votes
0
answers
65
views
Solving a differential equation related to the hypergeometric differential equation
I need to solve the following equation:
$x*(1 - s*x) y''[x] + y'[x] + r*y[x] =0,$
where $s$ and $r$ are two parameters.
It would seem that is similar to the hypergeometric differential equation, but ...
5
votes
0
answers
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views
Totally rational hypergeometric evaluations
This is a followup to the question at which rational points does the Hypergeometric function take rational values asked 10 years ago by
Eugene Starling, and is more a challenge than a question.
Let $F(...
1
vote
1
answer
151
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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:
$$
_2F_1(a,b;a+b;z)\sim -\frac{\Gamma(a+b)...
2
votes
0
answers
51
views
References for generalized confluent hypergeometric differential equation
According to Wolfram, a generalization of the confluent hypergeometric differential equation is given by:
$$y''+\left(\frac{2R}{x}+2F'+p\frac{H'}{H}-H'-\frac{H''}{H'}\right) y'+\left[\left(p\frac{H'}{...
4
votes
0
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Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$
I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. ...
11
votes
1
answer
475
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Integral representation of product of two Whittaker functions
Does anyone know anything about the following formula involving special functions:
$$\begin{multline*}
W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
7
votes
0
answers
307
views
Estimating the alternating sum $\sum_{j \ge 1} (-1)^j e^{-j^2} j^k$
I have been trying to get a lower bound on the following alternating sum but without much success:
$$
\sum_{j=1}^T (-1)^j e^{-j^2} j^k .
$$
For small values of $k$, this is easy because the first term ...
2
votes
1
answer
223
views
Asymptotic expansion of hypergeometric function ${}_3F_2$ for large parameters
I encountered the following hypergeometric function in my research: $${}_3F_2(2,1+n,1+n;1,2+n;z)$$ where $0<z<1$. I'm interested in its behavior for large $n$. Semilog plot suggests ...
2
votes
0
answers
132
views
Integral of Legendre's function
Is there any formula for computing the following integral $$\int_a^1(P^m_l)^2(x)\,dx \, ,\text{with} -1<a<1$$
where $P^m_l$ is the associated Legendre's function (of the first kind) of order $m\...
3
votes
1
answer
311
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Proving Clausen hypergeometric identity
How do I show the Clausen identity
$$
{}_2F_1\left(a, b; a+b+\frac{1}{2}; z\right)^2 = {}_3F_2\left(2a, 2b, a+b; a+b+\frac{1}{2}, 2a+2b, z\right)?
$$
I saw this on MathWorld but am unsure how to ...
11
votes
0
answers
625
views
What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?
The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions:
$\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...
0
votes
0
answers
91
views
Upper bounding the sum with hypergeometric and binomial probabilities
Could you please help me upper bound this tricky expression:
$$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$.
So far I only ...
3
votes
1
answer
277
views
Conformal mapping between two right-angled triangles
I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
17
votes
1
answer
859
views
Proof of certain $q$-identity for $q$-Catalan numbers
Let us use the standard notation for $q$-integers, $q$-binomials,
and the $q$-analog
$$
\operatorname{Cat}_q(n) := \frac{1}{[n+1]_q} \left[\matrix{2n \\ n}\right]_q.
$$
I want to prove that for all ...
2
votes
2
answers
418
views
A recurrence formula for the Legendre function $P_\mu^\nu(x)$
Im looking for a recurrence formula of type:
$$(\mu-\nu) x P_\mu^\nu(x) + P_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$
where $P_\mu^\nu(x)$ is the Legendre function of the first kind (solution to ...
3
votes
0
answers
39
views
Proving that a quotient of hypergeometric functions is smaller than a certain function
Im trying to prove that $\forall w \in (0,1), \forall k \in \left(0,\frac{1}{5}\right)$:
$$h_k(w) = \left[\frac{_2F_1\left(\frac{3}{2},1+\frac{1}{k};\frac{1}{2}+\frac{1}{k};\frac{1-w}{1+w} \right) }{...
2
votes
2
answers
229
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 $\...
0
votes
1
answer
502
views
Asymptotic expansion of hypergeometric 2F2
I would like to find an asymptotic expansion for the hypergeometric function
$$
_{2}F_{2}\left(a,b;c,d;z\right),\quad a,b,c,d\in\mathbb{R}.
$$
The parameters are fixed. $z$ is real and $z\rightarrow ...
3
votes
2
answers
369
views
Trying to bound the generalized hypergeometric function ${}_2F_3(x+1,x+1;1,1,1;\alpha)$ as $x\to \infty$?
(See also edit below)...
I am trying to get a nice, explicit, bound on the hypergeometric function
$$
{}_2F_3(a_1,a_2;b_1,b_2,b_3;\alpha),
$$
in the case of a large parameter. In particular I am ...
0
votes
2
answers
478
views
Sturm Liouville differential equation and hypergeometric functions
I'm trying to understand how to solve this differential equation:
$ [z^2(1-z)\dfrac{d^2}{dz} - z^2 \dfrac{d}{dz} - \lambda] f(z) = 0 $
I know the solution is related to the hypergeometric function ...
12
votes
1
answer
496
views
Yet another real-rooted polynomial
In this entry I asked for the real-rootedness of a polynomial, and two very interesting answers were given: one using Malo's theorem and the other a clever rewriting of the expression using Jacobi ...
5
votes
1
answer
549
views
Gegenbauer's addition theorem for Jacobi polynomials
I have the following identity,
$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z)
= 2 \, j_n(x) \, j_n(y) \;,$$
where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...