Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

Higher transcendental functions are frequently termed special functions. These functions were studied extensively in the eighteenth and nineteenth centuries-by Gauss, Euler, Abel, Jacobi, Weierstrass, Riemann, Hermite, Poincare, and other leading mathematicians of the day. Although many of the functions that they treated were quite recondite and are no longer of much interest today, others (such as the Riemann zeta function, the gamma function, and elliptic functions) are still intensively studied.

Before asking, please make sure that you define your notation very precisely, as
1. not everybody is familiar with the notation for special functions; and
2. a lot of special functions have different notational conventions, depending on the paper/book.

You might want to first check if the special function you are considering is discussed in Abramowitz and Stegun, the Digital Library of Mathematical Functions, or the Wolfram Functions site.

References:

https://en.wikipedia.org/wiki/Special_functions

https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms

"Special Functions and Their Applications" by R. Silverman

4626 questions
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Find a function $f_m $ that enumerates all subsets of $\{1,..,n\}$ of cardinality $m$ in ascending order

I'm Looking for a function $f_m: \{1,..,n\} \to \{(M\subseteq\{1,...,n\}: |M|=m\} $ that enumerates all subsets of $\{1,..,n\}$ of cardinality $m$ in ascending order. For example, for the set $\{1,2,3,4\}$ we'd have: $f_3(1)=\{ 1,2,3 \}, f_3(2)=…
Sudix
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Asymptotic form for a generalized hypergeometric function

${}_3{F_2}\left( {m+m_s,{m_s},{m_s};1 + {m_s},1 + {m_s}; - \frac{{{m_s}\overline \gamma }}{{m{\gamma _0}}}} \right)$, where $m$, $m_s$, $\gamma_0$, and $\overline \gamma$ are positive real numbers. When $\overline \gamma$ goes to infinity, can we…
Hui Zhao
  • 75
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Transfer foundamental functions into MeijerG function

How can we transfer $\frac{1}{{1 + \delta x}}G_{0,1}^{1,0}\left[ {\theta x\left| {_n^ - } \right.} \right]$ into the form like $G_{u,v}^{m,n}\left( {} \right)$, where $\sigma$, $x$, and $\theta$ are positive real numbers, and $n$ is a positive…
Hui Zhao
  • 75
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Is a Macdonald function a Bessel function with imaginary argument??

I mean that $$ K_{a} (x)= CJ_{a}(ix).$$ Here $C$ is a complex number, and $a$ is real. So is the Macdonald function a Bessel function in disguise (or proportional) of complex argument??
Jose Garcia
  • 8,506
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Meijer G-function

Can you please help me devise a series for the Meijer's G-function (i) with inceces m=3, n=0, p=1 and q=3, for a general real variable? The first difficulty that I am facing is the proper choice of an integration path, to use the residues' theorem.…
Yosdam
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Generalised Spherical Bessel Equation

can anyone give me a reference about the following kind of generalised spherical Bessel equation: $$\left[x^2\partial_x^2 +2x\partial_x +k^2(x^2+\alpha x)-l(l+1)\right]f(x)=0$$ or for…
m.a.137
  • 13
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An identity found on oblate spheroidal wave functions. Is it possible to prove this result?

For oblate spheroidal angular and radial functions as defined in Flammer's book: Spheroidal Wave Functions. I found an identity by numerical simulations shown that $$ \sum_{m=0}^\infty \sum_{n=m}^\infty\left[\frac{S_{mn}(-ic,\pm…
Jiaxin Zhong
  • 520
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What functions is this?

Good day everyone. I've encountered a rather strange function while looking up the asymptotic expansion of the confluent hyper geometric function in the NIST Handbook of Mathematical functions. Does anyone know what "env" means? Thanks!
anonymous
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Why the Fox H function is different?

Thank you in advance for your time and help. I am new with Fox H function. Allow me to ask some questions. why the definition is different between wiki and mathworld ? The links are below. You can see that the + and - sign are different in the…
Sunson29
  • 11
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weird bessel zero question

given 'a' and 'b' fixed i define the function $$ f(t)= bJ_{2t}(a) $$ here $ J_{n} $ is a Bessel function but in this cases i would be interested in getting the solutions (?? are there any ? ) for $$ J_{2t}(a)=0 $$ so for what 'index' 't' is the…
Jose Garcia
  • 8,506
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Dirac delta limit representation check?

It is well known that the Poisson kernel $$\eta_y(x)=\frac{1}{\pi}\frac{y}{x^2+y^2}$$ Provides a limit representation of the Dirac delta function $\lim_{y\to0}\eta_y(x)=\delta(x)$. In an attempt to verify this fact in Mathematica, I…
Kagaratsch
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Is there a special for the following function

Is there a special function that defines the following series $$T(x)=\sum_{n=0}^{\infty} e^{-a^2n^2}e^{-2x \pi n}$$ At first I tried the elliptic theta function.
Charles Hagwood
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Solving $x e^{-(x-\lambda)^2}=c$ for various $\lambda$

Suppose $\lambda$ and $c$ are real constants. I'm wondering if there's any special function that permits one to solve for the (real) variable $x$ in equations of the form $$x e^{-(x-\lambda)^2} = c$$ I'm familiar with the Lambert $W$ function and…
sourisse
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Meijer $G$ function and Lauricella Functions

As everyone know Meijer $G$ function is a vary general function that includes most of the known special functions; is it possible to consider Lauricella Functions as a special case of $G$?
Alessio Bocci
  • 187
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Does the following Gauss hypergeometric function increase with $x$

Given a function $f(x)={_2F_1}(-a,1;1-a;-x)$, where $00$. But I do not know how to prove it. Could you please help me to prove it or give me some hints? Many Thanks…
Dave
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