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Questions tagged [special-functions]

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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2answers
31 views

Equivalenece of all representations of $\exp$

I can name at least 4 different ways of representing $\exp$ function: Taylor series: For $x \in \mathbb{R}, \exp(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}$. Differential equation: $f: \mathbb{R} \to \...
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0answers
33 views

In interesting family of integrals related to the beta function

Consider the integral $$I(a,b)=\int_0^{\pi/2}\sin^at\ \cos^bt\ dt$$ As I have shown in numerous answers, $I$ has a close relationship with the beta function, namely $$I(a,b)=\frac12B\bigg(\frac{a+1}2,...
-1
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3answers
54 views

How to integrate $(e^x-1)/x$

How to integrate $\frac{(e^x-1)}{x}$ and find constant $c$ when $F(0)=0$. I have already tried wolframalpha, but this didn't help to solve this.
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0answers
23 views
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1answer
20 views

The generalized Laguerre polynomials: Are there any expressions valid for any case?

There are general expressions of the generalized Laguerre polynomials. For example: $$ L_n^{(\alpha)}(x) = {}_{n+\alpha}\mathrm{C}_n\ {}_1F_1(-n, \alpha+1, x), \hspace{20pt}(1) $$ $$ L_n^{(\alpha)}(x)...
1
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0answers
27 views

Tricomi confluent hypergeometric function

Let $$U(\alpha,\delta,z)=\frac{\Gamma(1-\delta)}{\Gamma(\alpha-\delta+1} {}_1\!F_1(\alpha,\delta,z) + z^{1-\delta}\frac{\Gamma(\delta-1)}{\Gamma(\delta)}\!F_1(\alpha-\delta+1, 2-\delta,z)$$ be the ...
0
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2answers
15 views

Continuous function that outputs even and odd numbers from natural inputs in a non-constant, non-alternating order?

For example, plugging 1, 2, 3, 4 into this function would produce results which are even, odd, odd, even, and that pattern would repeat. Is this possible for a continuous function? Can you make any ...
2
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1answer
33 views

Why do we only have one set of solutions for the PDEs of Legendre and Hermite polynomials?

This is an undergraduate-level mathematical physics problem. It may be trivial and basic to some of you, but it's important to me. In the mathematical physics course, the PDE for Hermite polynomials ...
2
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0answers
47 views

Series expansion of ratio of modified bessel functions

Let $I_{0}(z)$ and $I_{1}(z)$ be modified bessel functions of zero/first order (and first kind). How can I show that for the large value of $z$ it holds $$\frac{I_{1}(z)}{I_{0}(z)}\sim 1 - \frac{1}{...
2
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1answer
51 views

A connection between the confluent hypergeometric function and Bessel functions.

Let $b$ and $w$ be real parameters subject to $b\neq 1$. Let $x \in {\mathbb R}$. Define: \begin{equation} {\mathfrak N}(w,b):= \frac{2^{\frac{1}{2 (1-b)}+1} \left(\frac{1}{b-1}\right)^{\frac{1}{2 (b-...
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0answers
29 views

Do we have a formula to express $_3F_2(L+r-1+2q,1,r+2;L+r+2,r+3;1)$ in closed form?

Note: This is not a homework. I am interested to calculate $_3F_2(L+r-1+2q,1,r+2;L+r+2,r+3;1)$ where $L,r\geq 1$ are integers and $q\in [0,1]$ is a real number. In particular, Question: Do we ...
3
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1answer
50 views

How to evaluate the following sum if $n$ is an even number?

In a mathematical physical problem, the sum below needs to be calculated: $$ F_n(\xi) = \sum_{k=0}^{\operatorname{ceil} \left(\frac{n}{2}-1 \right)} (-1)^k \frac{\xi^{2k}}{n-2k} = \begin{cases} \...
-1
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1answer
29 views

Asymptotic behaviour of Floor function

What is the asymptotic behaviour for $[x]$ as $x \to \infty$ ($x \in \mathbb{R}$)? I want to have the sub-leading behaviour in $1/x$.
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2answers
298 views

How to calculate the Kampé de Fériet function?

This is a continuation of this post. The following is my original question in that post. Question: Is it possible to express $$\sum_{l=0}^\infty \sum_{r=0}^\infty\frac{\Gamma(L+r-2q)}{\Gamma(L+...
0
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0answers
22 views

Closed form of Harmonic series without digits 9 in its decimal representation

There is a question I've done before, that is In the Harmonic series $1+\frac12+\cdots+\frac1n+\cdots$, the fraction of $1/k$ is dropped if the decimal representation contains number digits 9, then ...
6
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2answers
329 views

Simplify $\sum_{l=0}^\infty \sum_{r=0}^\infty\frac{\Gamma(L+r-2q)}{\Gamma(L+r-1+2q)} \frac{\Gamma(L+r+l-1+2q)}{\Gamma(L+r+l+2)}\frac{r+1}{r+l+2}$

This question is a continuation of this post. Let $r,l,L\geq 1$ be integers. Assume that $q\in [0,1]$ is a real number. The authors obtained the following equation $36$ in their paper (I express in ...
2
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2answers
37 views

If $a,b,c$ are positive integers with $c\leq a+b,$ can I conclude that $_2F_1(a,b;c;1)$ diverges?

Recall that the hypegeometric series is defined by $$_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$$ where $z\in \mathbb{C}$ with $|z|<1$ and $(a)_n = a(a+1)...(a+n-1)...
5
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1answer
54 views

Limit of a Bessel function is a Gaussian

Let $x$ and $\nu$ be real. Is the following true? $$ e^{-\frac{x^2}{4}}=\lim_{\nu\rightarrow\infty}\Gamma(\nu+1)\left(\frac{2}{\sqrt\nu x}\right)^\nu J_\nu\left(\sqrt{\nu} x\right), $$ It appears to ...
3
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0answers
47 views

$\int_0^x\sin^a(t)\cos^b(t)dt=$?

I just thought of this integral: $$I(a,b,x)=\int_0^x\sin^a(t)\cos^b(t)dt$$ I know that $$I(a,b,\pi/2)=\frac{\Gamma(\frac{a+1}2)\Gamma(\frac{b+1}2)}{2\Gamma(\frac{a+b}2+1)}$$ And I know how to reach ...
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0answers
56 views

Can it be proved that the integral $I_1 = \frac{1}{2}$ iff $A=0$?

I have the following integral: $$ I_1= \int_{-\infty}^{\infty} \frac{d\tau}{2\pi i} \int_{-\infty}^{\infty} \frac{d\tau'}{2\pi i} \frac{1}{(\tau - i \epsilon)(\tau' - i\epsilon')}. M(\tau, \tau')$$ ...
1
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0answers
16 views

define an infinitely differentiable function that maps different sizes interval in same size intervals.

I have an interval split in two parts $I =[0, 1] = [0, th] \cup (th, 1] = I_1 \cup I_2$. I need to map the interval to a larger interval as: $f: [0,1] -> [0, 100]$ where: 1) $f(0)=0, f(1)=100, ...
2
votes
1answer
73 views

How do we prove that $\int_{0}^{\pi/2}\sin(t)^{2n+3}dt=\frac{4^n(2n+2)}{(2n+3)(2n+1){2n\choose n}}$?

I saw this integral in a paper on hypergeometric functions: $$S(n)=\int_{0}^{\pi/2}\sin(t)^{2n+3}dt=\frac{4^n(2n+2)}{(2n+3)(2n+1){2n\choose n}}\;\;\;\;\;\;\;\;\;\;\;(1)$$ I tried to prove it and got ...
2
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0answers
37 views

Algorithm for solving a large class of linear 2nd order ODEs with polynomial coefficients.

Let $\left\{ {\mathfrak P}_j \right\}_{j=0}^4$ be integers each of which is different from zero. Likewise let $p_1$,$p_2$,$q_1$ and $q_2$ be other integers that too are all different from zero. From ...
0
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0answers
21 views

The sum of infinite sum of modified Bessel function

I would like to ask a question about the following problem: $T(r,z)=\sum_{n=1}^{\infty}[C_{1n}I_0(\lambda_nr)+C_{2n}K_0(\lambda_nr)]cos(\lambda_nz)$ in which, $C_{1n}$ and $C_{2n}$ are coefficients, ...
1
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1answer
54 views

Repeated rescaling of differential equations

This is a follow-up question to Gauge transformation of differential equations. . As we know by repeatedly transforming the variables of a linear ODE we can generate a whole sequence of new ODE's ...
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0answers
13 views

Compute Infinite sum of Zero Order modified Bessel Function with COS

I am looking for the close integral series expression of following Modified Bessel Function. $T(r,z)=\sum_{i=0}^n[C_1I_0(\lambda_nr)+C_2K_0(\lambda_nr)]cos(\lambda_nz)$ I didn't find a closed ...
2
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1answer
44 views

On $s(\alpha)=\int_{0}^{\pi/2}\sin^{\alpha}(t)dt$

So, I am working on $$s(\alpha)=\int_{0}^{\pi/2}\sin^{\alpha}(t)dt$$ Looking for a general form. Although I am not (really) asking you to evaluate the integral, I have some questions about my methods. ...
2
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1answer
93 views

Solution of equation $y''+x^2y=0$

Does equation $y''+x^2y=0,$ where $y$ is function of $x$ have explicit solution? Perhaps with some conditions or in special case? I came across that when we have $x$ instead of $x^2,$ solutions are ...
1
vote
1answer
48 views

Finding the limit of an expression involving two Lerch transcendent functions

Consider the following expression: $$ f(s) = s^{n+1} \, \Phi(s^2,1,-1-\epsilon) - s^3 \Phi \left(s^2,1,-\frac{n}{2}-\epsilon \right) \, , $$ where $\Phi$ is the Lerch transcendent function (...
0
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1answer
40 views

$\int_{1}^{\infty}t^{k-n}dt=\frac{1}{n-k-1}$?

I am trying to find a series representation for $E_n(z)$, and I have gotten thus far: $$E_n(z)=\int_{1}^{\infty}\frac{e^{-zt}}{t^n}dt$$ $$E_n(z)=\int_{1}^{\infty}t^{-n}\sum_{k\geq0}\frac{(-z)^k}{k!}t^...
0
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1answer
46 views

Does this function have a specific name?

I came across this function as an example of a non-linear function being numerically fit to some data points using the Levenberg-Marquardt algorithm: $f(t; b_1, b_2, b_3) = b_1 (t+b_2)^{-\frac{1}{b_3}...
3
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1answer
75 views

Evaluating $\int_0^\infty (\operatorname{E}_n(x)e^x-\frac1{1+x})dx$

I want to evaluate $$I_n=\int_0^\infty \left(\operatorname{E}_n(x)e^x-\frac1{1+x}\right)dx=-\psi(n)$$where $\operatorname{E}_n$ denotes exponential integral and $\psi$ denotes polygamma function. ...
2
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0answers
42 views

Calculating Hermite Expansion Coefficents of $|x|$

I'm struggling to calculate the coefficents for the Hermite Expansion of the absolute value function and the indicator function $x \mapsto \mathbb{1}_{|x-u|\leq \delta}$ Background: I know, that for ...
1
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1answer
39 views

Mapping linear second order ODEs into each other.

By generalizing doraemonpaul's answer to Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$ I have the following result. Let $q_0(x)$, $q_1(x)$ and $q_2(x)$ by smooth functions (...
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2answers
46 views

Is there a closed form expression for the first zero of the first Bessel function?

$j_{1,1}$ denotes the first zero of the first Bessel function of the first kind. (That's a lot of firsts!) It's approximately equal to $3.83$. My question is, is there any closed form expression ...
1
vote
1answer
34 views

A Gauss's third order modular equation.

I would want a match for a typographical error (I think!). In a formula in the work of “CARL FRIEDRICH GAUSS, WERKE BAND III. GÖTTINGEN 1866” (www.archive.org). In chapter “ZUR THEORIE DER ...
1
vote
1answer
36 views

second order differential equation of special function

Bessel function $J_n(x)$ and $Y_n(x)$ obeys the following differential equation: $x^2 y''(x)+x y'(x)+(x^2-n^2)y=0,$ where superscript ' denotes differentiation with respect to $x$. In general, ...
0
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0answers
71 views

Evaluating $\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx$

According to this post $$Q=\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx=\frac{1}{8}\zeta \left( 4 \right) - \frac{1}{2}\zeta \left( 2 \right){\ln ^2}(2) + \frac{1}{{12}}{\ln ^4}(...
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0answers
17 views

Orthogonal polynomials as eigenfunctions of a second-order difference operator

I am Reading the theorem 6.1.3 of this book https://books.google.com.mx/books?id=RusIDAAAQBAJ&pg=PA146&lpg=PA146&dq=up+to+normalization,+the+charlier,+krawtchouk,+meixner,+and+chebyshev%E2%...
2
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1answer
50 views

Is there any special function corresponding to $\int_{-1}^1 \frac{1-e^{b(x-a)}}{(x-a)^2}\sqrt{1-x^2}dx$?

I try to get an expression for this difficult integral: $$\int_{-1}^1 \frac{1-e^{b(x-a)}}{(x-a)^2}\sqrt{1-x^2}dx\quad b>0,\;a>1$$ It could also be written in terms of trigonometric functions ...
2
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1answer
82 views

Is it possible to simplify $\psi^{(2)}(\frac18)$ or $\psi^{(2)}(\frac pq)$?

Is it possible to simplify $\psi^{(2)}(\frac18)$, where $\psi$ denotes the polygamma function? Or more generalized, $\psi^{(2)}(\frac pq)$ and $\psi^{(2n)}(\frac pq)$? Background Noticing there is ...
4
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0answers
52 views

Exact value of Elliptic Integrals.

I was taking currently in a elementary calculus course where i found how to find arc lengths of a smooth continuous curve. so here is how i started : $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow y=...
1
vote
1answer
37 views

Asymptotic behavior of $_3F_2$ at unit argument for large values of parameters

I have to deal a lot with functions which are $_3F_2$ with unit argument, and I need to find their behavior for large values of the parameters. One example is $$_3F_2(n,n,n;2n,n+x;1)$$ where I'm ...
0
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0answers
24 views

Derivation of Meijer-G function with respect to parameters

I am working on my research, and I have one optimization problem to be given as: $\frac{d}{d \theta}G^{\,2,1}_{1,3}\left( \begin{array}{c} 1\\ m(\theta),m(\theta),0 \end{array} \middle\vert\ Z(\theta)...
3
votes
0answers
36 views

Building solutions to forth order ODEs out of products of solutions to second order ODEs.

Let $r$, $A$, $A_0$,$A_1$ and $A_2$ be real numbers. By generalizing the approach given in my answer to How can I solve the following higher order ODE? I have found the following. Define the ...
0
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0answers
34 views

Extreme-mega-ultra-crazy hypergeometric functions

I've been lusting over hypergeometric functions, and came up with some questions. Here goes. I've defined the following functions, and I want to know if there are any closed forms for them. $$f_1(2;...
1
vote
0answers
37 views

Closed form for $_2 F_2(\alpha,3\alpha;2\alpha,4\alpha;x)$

I came up with a potentially interesting hypergeometric function. I know that $$_2 F_2(\alpha,3\alpha;2\alpha,4\alpha;z)=\sum_{n=0}^{\infty}\frac{(\alpha)_n(3\alpha)_n}{(2\alpha)_n(4\alpha)_n}\frac{z^...
3
votes
0answers
91 views

Solutions of $f(x)=\prod_{k=1}^n f^{(k)}(x)$

A while back I had wondered about solutions to the differential equation $$f(x)=\prod_{k=1}^\infty f^{(k)}(x),\tag1$$ and it became clear that in order for the product to converge, the derivatives ...
1
vote
2answers
55 views

An integration related to incomplete gamma function

I have no clues about the following equation, expecting some help from anyone. $\int_\beta^\infty e^{-x \theta}\frac{1}{\theta}(\frac{\theta}{\beta}-1)^{-\alpha}d \theta=\Gamma(1-\alpha)\Gamma(\alpha,\...
6
votes
0answers
111 views

Evaluation of $\int_0^1\frac{\ln(x)\ln(x+1)\ln(x^2+x+1)}{(1-x)(1+x^2)}dx$

I originally saw this logarithmic integral pop up on the Integrals and Series forum but due to the inactivity there no conversation has developed. I originally tried generalizing the integral to $$I(...