# 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.

840
questions

0
votes

1
answer

17
views

### Prove that the regularized incomplete beta function monotone with each of its parameter

Consider the regularized incomplete beta function $I_x(a, b)$ with $x \in [0,1]$ and $a, b > 0$. I am hypothesizing that the function is monotone decreasing with respect to $a$ and monotone ...

2
votes

1
answer

69
views

### Proof of the monotonicity of a regularized incomplete beta function

I want to prove the monotonicity of $I_r(nr, 2+(1-r)n)$ on $n$ but has no clues. The $I$ is the regularized incomplete beta function, defined as follows:
$$I_r(nr, 2+(1-r)n)=\frac{\int_0^r x^{nr-1}(1-...

-1
votes

0
answers

77
views

### Integral similar to the incomplete gamma function

Let us consider the integral
$$
I:=\int\frac{e^{-x}(a+bx)^n}{x}\,dx,
$$
where $a,\,b\in\mathbb{R}$, and $n\in\mathbb{N}$.
Does there exist any expressions of this integral via incomplete gamma ...

2
votes

1
answer

106
views

### Do Zernike polynomials form an orthogonal basis of $L^2 ( \mathbb{D} )$?

The family of Zernike polynomials is defined as follow over the unit disc $\mathbb{D}= \{ x \in \mathbb{R}^2, \ \lVert x \rVert \leq 1 \}$.
For $n \geq 0$ and $0 \leq m \leq n$ such that $n-m$ is even,...

4
votes

1
answer

153
views

### Uniform decay of $J'_{\nu}(x)$ for $x\gg1$

I need a uniform decay estimate for the derivative $J'_{\nu}(x)$
of the Bessel functions.
By `uniform' I mean an estimate independent of $\nu$, at least for a range of orders like $\nu\ge0$.
For $J_{\...

0
votes

1
answer

60
views

### PDF of the difference of two Beta Prime distribution

I am struggling to find the PDF of the difference of two Beta Prime distribution.
Definition
A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...

1
vote

0
answers

43
views

### $ \lambda^{*}(n) $ minimal polynomials

I already asked a closely related question on MO, but didn't received any answer.
Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here.
Is ...

0
votes

0
answers

27
views

### Finding a closed form of the following infinite summation of product of bessel functions

I have asked the same question to math stack exchange, but couldn't get an answer yet. So, I thought maybe it is a good idea to share to here.
While doing my research, I encountered the following ...

6
votes

1
answer

476
views

### A functional equation

I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...

1
vote

1
answer

111
views

### Radial Fourier transform vs Hankel transform

I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions.
Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. ...

4
votes

0
answers

138
views

### Uniform bound for Bessel functions

In NIST the following bound is claimed: $|J_\nu(x)|\le 1$ for all $x,\nu\ge0$. This is trivial for integer $\nu$, and it is pretty easy to prove a bound with 1 replaced, say, by 2. Does anyone have a ...

1
vote

0
answers

90
views

### Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$

I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...

4
votes

2
answers

670
views

### Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$

Let's state with $\psi^{(1)}$ the trigamma. Calculate the order:
$$\mathcal{S} = \sum_{n=1}^\infty (-1)^{n-1} (\psi^{(1)}(n))^2$$
(Cornel Ioan Valean)
I uploaded this question here https://math....

1
vote

0
answers

64
views

### Fitting a product into the quintuple or Jacobi triple product

The Rogers-Ramanujan functions fit nicely into the QPI or JTP. In fact we have that $$(q^{5};q^{5})_{\infty}(q,q^{4};q^{5})_{\infty}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(5n^{2}-3n)}{2}}$$ and we ...

5
votes

1
answer

338
views

### Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE
Certainly, I apologize for any oversight. Here's a more refined ...

5
votes

2
answers

3k
views

### Nicer expression for 2.1369288...?

In Drift Analysis and Evolutionary Algorithms Revisited by Johannes Lengler and Angelika Steger in Theorem 10, there is mention of a constant "$2.2$", and in the proof it becomes apparent ...

0
votes

0
answers

67
views

### How to write Tricomi's confluent hypergeometric function in terms of Meijer-G function

I am calculating a closed form expectation and I encountered the Tricomi's confluent hypergeometric function
(aka confluent hypergeometric function of the second kind) given by integral $U\left( a,b,z ...

2
votes

1
answer

168
views

### Has the function $F_s(x)=\sum_{k=0}^{\infty}\frac{x^k}{\Gamma(k+1)^s}$ been studied before?

While studying an application of Grönwall's inequality I found that the function
$$
F_s(x)=\sum_{k=0}^{\infty}\frac{x^k}{\Gamma(k+1)^s}
$$
for $s\geq0$ in some cases provides a sharper bound.
I had a ...

1
vote

1
answer

170
views

### Green's function for a linear PDE initial value problem

For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem
$$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...

6
votes

1
answer

165
views

### About the high-order derivatives of Lambert function

In the mid seventies, in my former research group, we found that the $n^{\text{th}}$ derivative of $W_0(x)$ could write
$$\frac {d^n\,W_0(x)}{dx^n}=(-1)^{n+1}\,\,\frac{\,P_n(w)}{ e^{nw}\,(1+w)^{2n-1}}\...

1
vote

1
answer

208
views

### Antiderivative of Meijer G-function

In the python sympy CAS framework one strategy to compute integrals is to transform the integration kernel to a MeijerG-function, obtain the corresponding antiderivative as MeijerG-function and ...

3
votes

2
answers

371
views

### Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by
\begin{equation*}
\frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...

14
votes

1
answer

678
views

### Power series $x^n/n$ with one plus, then two minuses, then three plusses, and so on

Which function is represented by the powerseries
$$1-\frac{x}{2}-\frac{x^2}{3}+\frac{x^3}{4}+\frac{x^4}{5}+\frac{x^5}{6}-\frac{x^6}{7}-\frac{x^7}{8}-\frac{x^8}{9}-\frac{x^9}{10} + \dots$$
(one plus, ...

2
votes

1
answer

124
views

### How to calculate this integral of squared Tricomi hypergeometric function

How to solve this integral
$$
\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r
$$
where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...

22
votes

2
answers

2k
views

### Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?

Is it known whether $$\int_0^1 e^{-x^2} \, dx$$ is irrational? It is well-known that $\int_0^\infty e^{-x^2} \, dx=\frac{\sqrt{\pi}}{2}$ which is irrational, but what about the prior integral? Also, I ...

1
vote

1
answer

167
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 $m$...

0
votes

1
answer

154
views

### Asymptotic behavior of the polylogarithm function and generalisation

So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper:
$$
\sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...

4
votes

1
answer

104
views

### Eigenvalues of the modified Mathieu equation with normalizable solution

The Mathieu equation (https://en.wikipedia.org/wiki/Mathieu_function) is $y''+(a-2q\cos(2z))y=0$. The modified Mathieu equation is obtained by replacing $z$ with $\pm iz$:
$$y''-(a-2q\cosh(2z))y=0.$$
...

-1
votes

1
answer

131
views

### Is there a single function describing this piecewise function?

I am examining the piecewise function given by the following. $f(x) = \pi - \arctan(\frac{2x}{1-x^2})$ when $0 \leq x < 1$, $f(x) = \frac{\pi}{2}$ when $x=1$, and $f(x) = \arctan(\frac{2x}{x^2-1})$ ...

6
votes

1
answer

373
views

### When are the chirp signals orthogonal?

Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...

4
votes

1
answer

280
views

### Double q-analog of Pochhammer

Has the function
$$(z;q_1,q_2)_\infty := \prod_{n_1,n_2=0}^\infty (1-z \, q_1^{n_1} q_2^{n_2}), \quad |q_1|,|q_2|<1$$
been studied in the math literature? For example, does it obey any difference ...

1
vote

2
answers

355
views

### A closed formula for a sum involving hypergeometric function

Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !...

3
votes

1
answer

333
views

### How to find partial derivatives of the Beta Function?

I was reading the book (Almost) Impossible Integrals, Sums and Series. The author used a method involving taking partial derivatives of the Beta Function to solve some integrals.
$$B(x,y)=\int_0^1u^{x-...

1
vote

1
answer

177
views

### T functions arising from derivatives of incomplete Gamma function

Here the derivatives of the incomplete gamma functions are described via:
$$
T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right|...

2
votes

1
answer

201
views

### Sum of Bessel function with integer parameters and fixed argument

Question.
Let $J_{\nu}$ be a standart Bessel function of the first order. What is the asymptotic of the sum $\sum_{n\ge 0}|J_n(t)|$ as $t\to\infty$? An upper bounds stronger than $O(t)$ are also ...

1
vote

0
answers

75
views

### How to calculate the Integral with confluent hypergeometric function

How to prove this.Thank you in advance
Let $\delta,\beta>0$ How to prove this
\begin{align}
& \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...

9
votes

1
answer

633
views

### What is the value of $j(2\sqrt{-163})$?

My question is how to calculate the value of $j(2\sqrt{-163})$ and its minimal polynomial, where the $j$ is elliptic modular function (see https://mathworld.wolfram.com/j-Function.html). The class ...

3
votes

2
answers

399
views

### Functional equations based on composition

I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...

0
votes

0
answers

153
views

### How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?

The classical Euler gamma function can be defined by the integral
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0.
\end{equation*}
Its ...

0
votes

1
answer

129
views

### Is $\Gamma(z,1)\not=0$ for all $z$ with $\Re(z)<0$?

I found this paper online which appears to present zeros of the incomplete gamma function within the right half plane. It makes me think that there are no zeros in the left half plane. Not sure how to ...

5
votes

0
answers

115
views

### Ratio of theta functions as roots of polynomials

I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\...

7
votes

0
answers

217
views

### Analytic continuation of Dixon's identity

Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...

0
votes

0
answers

36
views

### Bessel functions of matrix argument in the scalar case

Herz (1955) provides the following equality:
$$
A_\delta(-\lambda) - A_\delta(-\lambda)\lambda^{-\delta} = -\sin(\pi\delta)B_\delta(\lambda)/\pi
$$
where $A_\delta$ and $B_\delta$ are the Bessel ...

5
votes

0
answers

242
views

### Questions on Gauss's geometric interpretation of spherical functions

(This question was initially posted on HSM stackexchange, but eventually I came to conclusion that it is too mathematical to be answered there.)
In the physics chapter of his biography of Gauss, W.K. ...

3
votes

0
answers

161
views

### Hilbert's 13th Problem and series solutions for the reduced sextic, septic, and octic?

I. Reduced equations
One can eliminate 3 terms from the general quintic, sextic, septic, and octic using a Bring-Jerrard transformation to get the reduced forms in radicals,
$$x^5+(x+p) = 0$$
$$x^6+(x+...

13
votes

6
answers

2k
views

### Closed form of an infinite series

Does the following infinite series have a closed form?
$$
\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}
$$

2
votes

0
answers

212
views

### Zeta function associated with a function $f$

Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define
$$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt.
$$
Is there a general formula that ...

7
votes

2
answers

521
views

### Weak convergence related to Hermite polynomial?

I am reading Griffiths's quantum mechanics book; in the section about harmonic oscillators, he wrote out the amplitude of wave function, and compared with the classical harmonic oscillators. He ...

2
votes

4
answers

675
views

### Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?

Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function?
How about the positivity, monotonicity, and convexity of the ...

2
votes

1
answer

202
views

### An integral transform computation

In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2.
they note that
\begin{align}
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds
= 2^{-\nu/2} \pi^{-...