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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Subtraction of two similar Meijer G-functions

Can we get another Meijer G-function from the following subtraction? $$ G^{0, 5}_{5, 2}\left( z \left| \begin{matrix} (1, a, b, c, d) \\ (e, f) \end{matrix} \right.\right) -G^{0, 5}_{5, 2}\left( z \left| \begin{matrix} (\frac{1}{2}, a, b, c, d)…
Dante
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Inverse of $f(x)=x/\log(1+x)$ for $x>1$

Is there an expression of $f^{-1}$ in terms of product-log or other special functions which holds for $x>1$? WolframAlpha finds the solution: $$g(y)=-y\,W\big(-\mathrm e^{-1/y}/y\big)-1,$$ which however only holds for $-1
AndreA
  • 201
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${\rm E}_n$ function for non-integer $n$

The ${\rm E}_n$ function implementation in scipy is valid only for positive integer values of $n$, but I have an expression with real $n$ to evaluate. Is there an expression for ${\rm E}_n$ in terms of other special functions (preferably implemented…
Kyle
  • 213
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Is this function "$h$" symmetric of the plane $x=y$?

$h=\left\{\begin{matrix} f,x
Ian
  • 1,391
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Chebyshev Polynomials and the Hypergeometric Function

A problem related to Chebyshev Polynomials and the Hypergeometric Function involves transformation from one function to another. The task is to transform the Chebyshev polynomial into its correct scaled Hypergeometric Function. $$ T_{2n} (x) = ( -…
Lawrence Calixto
  • 11
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Special Functions prerequisites

I want to study special functions for the first time and I've seen that S. F. sometimes are solutions for differential equations. Do I have to learn certain topics other than calculus before studying S. F. for the first time?
Undergraduate Student
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Find value of a Fox H-function

I tried to evaluate the values of Fox H-function $H_{2,2}^{1,1}(t)$ at point $t>0$. We have $$ H_{2,2}^{1,1}\left[ t\left| \begin{array}{cc} (0,1) & (1,1)\newline (1,1) & (\alpha,1) \end{array} \right. \right] = \frac1{2\pi i} \int_L…
pabodu
  • 435
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How does $\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x=1+\frac{x(x-1)}{2j^2}+O\left(\frac{1}{j^3}\right)$

How does $$\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x=1+\frac{x(x-1)}{2j^2}+O\left(\frac{1}{j^3}\right)$$ my attempt: It looks like maybe the taylor series of the numerator at 1 times the geometric of the…
user5389726598465
  • 1,171
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Generalised hypergeometric function $_4F_3$ at unit argument

Is there a way to simplify the following generalised hypergeometric function evaluated at unit argument $$ \, _4F_3\left(-n,3+n,\frac{2}{3},\frac{5}{3};2,-\frac{1}{3}-n,\frac{8}{3}+n;1\right) $$ with $n\geq0$ being an integer?
user12588
  • 399
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problem about the $\mathscr{E}_1(z)$

we know that: $$ \mathscr{E}_1(z)=\int_z^{+\infty}\frac{e^{-t}}{t}\mathrm{d}t,\quad |\arg(z)|<\pi $$ my question is that : why the $|\mathrm{arg}(z)|<\pi$
我心永恒
  • 45
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Realization of Bessel functions

As I know when whe trying to execute Bessel functions when $z < 8$ we using this formula $$J_\nu(z) = (\frac{1}{2}z)^\nu \sum_{k = 0}^{\infty} \frac{(-\frac{1}{4}z^2)^k}{k!\Gamma(\nu+k+1)}$$ And thats formula I found when $z \geq 8$ $$J_\nu(z)…
Danil Gholtsman
  • 243
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How to expand $e^{i k \frac{1}{1 + \mathrm{cos}(\omega t) + \mathrm{cos}(2 \omega t)}}$??

As Jacobi-Anger expansion suggest: $e^{i z \mathrm{cos}(\theta)} = \sum_{n=-\infty}^{\infty} i^n J_n(z) e^{i n \theta}$ What if $\mathrm{cos}(\theta)$ from the expression above would be replaced with some other function $f(\theta)$. Is there a…
slm992
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An incomplete double series: Is there a special functions representation?

Consider the sum $$Q(v,u) = \sum_{l=0}^\infty\sum_{k=0}^{l}\frac{u^l}{l!} \frac{v^k}{k!}$$ which arises from the inverse Laplace transform of $f(s) = \frac{1}{s(s-a)}e^{b/s}.$ Is there a means to express $Q(v,u)$ in terms of some special…
kevinkayaks
  • 1,444
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Integral of the fractional part of $\frac1x$ multiplied by $x$ on interval $(a,b), a\ge 0$.

I'm interested in finding the value of the integral of $\left\{\frac{1}{x}\right\}\cdot x$ (the fractional part of $\dfrac{1}{x}$ multiplied by $x$) on the interval $(a,b), a\ge 0$ the integral of $\left\{\frac{1}{x}\right\}$ (fractional part of…
user9532
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Looking for an unbounded, monotonic, non-symmetric s-shaped function

I am hoping to find a function that corresponds to an s-shaped curve that satisfies the following properties: 1. unbounded 2. it is not symmetric about its inflection point 3. its derivative at the inflection point is not infinity 4. it is…
BillB
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