Questions tagged [special-functions]
Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
829
questions
21
votes
1
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A multiple integral that seems related to the $\zeta$ function at even integers
I came across this integral that seems related to the Riemann zeta function $\zeta(2n)$ evaluated at even integers $2n \in 2\mathbb{Z}$. Letting $n$ be an even integer, define the multiple integral ...
0
votes
1
answer
448
views
When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?
I am studying properties of the two-parameter Mittag-Leffler function.
$$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and ...
0
votes
1
answer
160
views
Defining Legendre polynomials in terms of a sinusoidal function for $|x| \leq 1$
Would it be possible to define Legendre polynomials in terms of a sinusoidal function for $|x|\leq 1$ in a similar manner to Chebyshev polynomials being defined as $T_n(x) = \cos(n \cos^{-1}(x))$?
...
0
votes
0
answers
142
views
Number of open tours by a biased rook on a specific $f(n)\times 1$ board which end on a $k$-th cell from the right
We have a simple structure - biased rook of the two types.
Biased rook of the first kind which make open tours on a specific $f(n)\times 1$ board where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ ...
3
votes
0
answers
154
views
On analogues of Weber's formula
Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that
$$
\int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta})....
3
votes
1
answer
195
views
6-j symbols and hypergeometric series
What’s the correct formula for $_{4}F_{3}(a,b,c,d;e,f,g;1)$ where $a+b+c+d-e-f-g=-1$?
The Wolfram Alpha formula involves $6j$ symbols and makes no sense for some specific cases. For example, $_{4}F_{...
1
vote
0
answers
98
views
Asymptotic behavior of hypergeometric function ${}_{3}F_{2}(a,b-n,c-n;d-n,e-n;1)$ for $n\to\infty$
Suppose $a,b,c,d,e\in\mathbb{R}$ are such that $d+e-a-b-c>0$ and $d,e\notin\mathbb{Z}$. I would like to know whether it is possible to deduce an asymptotic formula for the sequence given by the ...
1
vote
1
answer
295
views
How to prove the convexity of a simple function involving a ratio of two polygamma functions?
Let
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0
\end{equation*}
and
$$
\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.
$$
In the literature, ...
1
vote
0
answers
74
views
On the properties of the even and odd components of the function $f_n(x)=(n-1) \text{Li}_n\left(e^x\right)-x \text{Li}_{n-1}\left(e^x\right)$
Let's define
$$f_n(x)=(n-1)\operatorname{Li}_n\left(e^x\right)-x\operatorname{Li}_{n-1}\left(e^x\right)$$
$$feven_n(x)=\frac{f(x)+f(-x)}2$$
$$fodd_n(x)=\frac{f(x)-f(-x)}2$$
At even $n$ the even part ...
2
votes
1
answer
426
views
This equality numerically looks well. Is there any justification?
Is it true that
$$f(x)=\lim_{n\to\infty} 2 \sum _{k=0}^n \left((k-1) \text{Li}_k\left(\frac{f(x)}{n^2}\right)-x \text{Li}_{k-1}\left(\frac{f(x)}{n^2}\right)\right)?$$
Here, $f(x)$ is an arbitrary ...
3
votes
1
answer
385
views
Asymptotic behavior of infinite product of cosines
Consider the function
$$F(z) := \cos(z) \cos(z/3) \cos(z/5) \cos(z/7) \cdots$$
Note that $\cos(z/n) = 1 - \frac{z^2}{n^2} + \cdots$ so the product is absolutely convergent to an entire function.
I ...
2
votes
1
answer
298
views
Two questions about an integral involving double product of Bessel functions
Let us define the following integral :
$$W_n(r)=r\int_0^{+\infty} J_1(rt)[J_0(t)]^n dt,$$
with $r>0$ a real number and $n\in\mathbb{N}$ and where $J_0(x)$ and $J_1(x)$ are Bessel functions of the ...
5
votes
1
answer
275
views
Relationship between Lambert $W$ function and Hypergeometric function
The Lambert $W$ Function is defined in this Wikipedia entry, while the Hypergeometric Function is defined in this other Wikipedia entry. There exists also a multivariate generalization which solves ...
8
votes
2
answers
1k
views
Question about functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $x$ is prime whenever $f(x)$ is prime
Let $f: \mathbb{\mathbb{Z}^+} \to \mathbb{Z^+}$ be a function and suppose
$(\star)$ For all integers $x \geq 3$, if $f(x)$ is prime, then $x$ is prime.
A trivial example of such a function is the ...
2
votes
0
answers
71
views
Uniform bound on a certain family of hypergeometric functions
We have the following problem, which we can't solve.
Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
1
vote
1
answer
250
views
Asymptotic behavior of an ODE
Consider the following ODE eigenproblem of $y(x)$
\begin{equation}
y'' + [\varepsilon + b^2 x - (a + \frac{b^2}{2}x^2)^2 ] y=0
\end{equation}
with eigenvalue $\varepsilon$, real constants $a,b$. ...
9
votes
1
answer
638
views
$\mathrm{Li}(x)$ vs $x/\log x$
I need an explicit lower bound for $\mathrm{Li}(x)$ in terms of $x$ and $\log x$. Say, Wikipedia gives
$$
\mathrm{li}(x) >\frac x{\log x}+\frac x{(\log x)^2}
$$
for $x>e^{11}$, see the ...
2
votes
2
answers
133
views
Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type
Let $J_\nu$ be a Bessel function of the first kind and let $\{\lambda_{n, \nu}\}_{n\ge 1}$ be a sequence of its zeroes. I claim that
$$
\inf_{n\ge 1}\bigg|\sqrt{\lambda_{n,\nu}} J_{\nu+1}(\lambda_{n,\...
12
votes
2
answers
1k
views
Does there exist an upper bound on the Fourier coefficients of the reciprocal theta function $\frac {1}{\theta}$?
Define the theta function as
$$
\theta(x) = \sum_{n=-\infty}^\infty e^{-\gamma(x+n)^2}
$$
where $\gamma>0$. Clearly, $\theta$ is 1-periodic, non-zero and smooth. Therefore, the reciprocal map $x \...
7
votes
1
answer
489
views
continued fraction for logarithmic integral
Does the logarithmic integral function $\operatorname{li}(x)$ have the continued fraction expansion
$$\operatorname{li}(x) = \cfrac{x}{\log x -1 -{}} \ \cfrac{1}{\log x -3 -{}} \ \cfrac{4}{\log x -...
0
votes
1
answer
264
views
Laplace transform and Laguerre Polynomials
What is the kernel $K(t)$ of the following Laplace transform equation:
$$\int_{0}^{+\infty}e^{-(x+y)t} K(t) dt= \sum_{n=0}^{\infty}\varphi_{n}^{\alpha}(x)\varphi_{n}^{\alpha}(y),$$
where $\varphi_{n}^{...
3
votes
1
answer
170
views
On integral representation of Whittaker $W$ functions
According to NIST, the integral representation of Whittaker $W$ functions
$$
W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(%
\frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
4
votes
1
answer
447
views
Beta function, harmonic numbers, and integral values
A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k
$$
where $\beta_x( -1 - ...
2
votes
1
answer
172
views
Generalized Selberg integral
I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions
$$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\...
1
vote
1
answer
110
views
Is trace of a slice of an elementary function of a matrix also elementary?
Let we have an elementary function $f(W)$, applicable to a matrix.
Now consider the function
$g(x)=\operatorname{tr} f(W+x),$
where $x$ is scalar. Is $g(x)$ necessarily an elementary function?
Simple ...
3
votes
0
answers
114
views
Asymptotics of a combinatorial series
I am interested in the exact asymptotics of the following combinatorial series (which arises from the study of a Markov chain):
$$F(q):=\sum_{k \ge 1} \frac{q^{k^2}}{(q;q)_k^2}\quad \mbox{as } q \to 1^...
5
votes
0
answers
224
views
Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
3
votes
2
answers
69
views
Optimal scaling of Lipschitz estimates in generalized geometric series
If we did not know it before, then wikipedia teaches us the generalized geometric series
$$\sum_{n \ge 0} \binom{n+k}{n} (1-\mu)^{n} \mu^k = \frac{1}{\mu}.$$
We can then study for $0 <\varepsilon &...
3
votes
0
answers
207
views
Does there exist a type of discriminant not only for irreducible polynomials but also for exponential functions, logarithm functions?
I think discriminant is the strongest tool that I've used_ https://math.stackexchange.com/q/4035405/822157, however, does there exist a type of discriminant not only for irreducible polynomials but ...
3
votes
0
answers
229
views
definite integral with incomplete gamma function and exponential
While working with electron density computations in quantum chemistry, I encountered the following improper integral:
$$
I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
3
votes
1
answer
175
views
Analytic or holomorphic extension of the ellipse perimeter function
Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$.
Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^...
3
votes
1
answer
113
views
General formula for the integral w.r.t to Marchenko-Pastur density, of the ratio of degree $\le 2$ polynomials
Question. Is there a closed-form formula (via standard objects like rational functions, radicals, special functions, special numbers like Catalan numbers, etc.) expressing integrals of rational ...
9
votes
0
answers
318
views
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 ...
1
vote
1
answer
212
views
Sum of negative roots of a $5^{th}$ degree monic polynomial
Let $f(x)$ be a $4^{th}$ degree monic polynomial say $f(x) = x^4 + a_1x^3+a_2x^2+a_3x+a_4$ with the property that $a_1<0, a_4>0$ and $a_2<a_3$. They by Descartes' rule of signs we can ...
0
votes
0
answers
69
views
Integration of fractional function over Rice distribution
Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as
\begin{equation}
\int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx
\end{equation}...
1
vote
0
answers
118
views
A series with zeroes of Bessel functions
Consider a finite sum
$$
S_n(t)=\sum_{m=1}^n \frac{J_\nu(z_{m,\nu} t)}{J_{\nu+1}(z_{m,\nu})}, \nu>0, 0\leq t <1,
$$
$z_{m,\nu}$ are ordered real positive zeroes of the Bessel function $J_\nu(t)$....
1
vote
2
answers
679
views
Inverse of exponential integral function
The exponential integral function $x \mapsto E_1(x)$ is strictly decreasing on the positive real
axis and, so, is globally real analytically invertible there. Where can I find information concerning
...
1
vote
1
answer
168
views
Taylor expansion of Modified Mathieu functions
Do we know the Taylor expansion at $0$ of the radial Mathieu functions $(\mathsf{Mc}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 0}$ and $(\mathsf{Ms}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 1}$, for $q \in \...
0
votes
0
answers
115
views
Can the Bessel functions tend to a plane wave?
Can the Bessel functions tend to a plane wave?
If I have this function:
$$
y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)
$$
...
2
votes
0
answers
56
views
Class of differentiated Gamma functions: are there any algebras where they are elementary?
There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function.
They include the Gamma function, Polygamma function, and Hurwitz Zeta ...
1
vote
0
answers
57
views
Inequalities in special function cones
We consider the Banach space $X=C([0,1])$ endowed with the norm $\|v\|_{\infty}=\max _{t \in[0,1]}|v(t)|$ and, we define the cone
$\mathcal{C}=\{u \in X \mid u \mbox{ is concave, } u \geq 0, u(0)=u(1)=...
3
votes
1
answer
182
views
Can the following sum be counted or expressed in terms of special functions?
Let us define this sum as a function of $z \in \mathbb{C}$ with some positive parameter $a$
$$
f(z; a) = \sum\limits_{n = 0}^{\infty}\frac{|z|^{2n}}{n!}e^{-ian^2}.
$$
Probably, it can be expressed (or ...
0
votes
0
answers
48
views
Is it possible that in certain rings the power series representing special functions are expressable via series representing elementary functions?
Let's consider the Taylor power series of a function on real numbers.
Some of them represent elementary functions, and some of them represent special functions. The special functions cannot be ...
1
vote
0
answers
74
views
Complex integration related to finite temperature number density correlation function of 1d free fermion
I am looking for an explicit formula for this complex integral.
$$\oint_C \frac{d z}{2 \pi i} \frac{z^{-(x+1)}}{1+e^{-\beta(z+1 / z-\mu)}},$$
where $x\in \mathbb{Z},\ \beta,\mu\in \mathbb{R}\ $. The ...
0
votes
1
answer
760
views
Error function of multivariate Gaussian
I'm trying to prove that a sequence of functions $(k_N)_{N\in\mathbb{N}}$
$$k_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$
where $r>0$, $k>2$ and
Edit: I have forgot to ...
7
votes
4
answers
491
views
Is this closed-form summation a special case of known Lerch zeta function formulas?
With some Poisson summation manipulations (credit: Michał Pacholski) I have convinced myself of a closed form expression for this conditionally convergent series:
$$\sum_{n=-\infty}^\infty \frac{e^{in\...
8
votes
0
answers
245
views
Variations on Gauss' trick
Cross-posted from MSE. This question is inspired by these two:
Non-trivial values of error function erf(x)?
Where is the mass of a hypercube?
Upon reading these two, I realized there might be a ...
1
vote
1
answer
151
views
Higher-order asymptotics of generalized hypergeometric function
I have a question about higher-order asymptotics of generalized hypergeometric functions. According to https://dlmf.nist.gov/15.4
the following is well known:
$$
_2F_1(a,b;a+b;z)\sim -\frac{\Gamma(a+b)...
8
votes
2
answers
995
views
An interesting infinite product involving the factorial function with connection to the K and gamma function
I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. ...
4
votes
0
answers
186
views
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. ...