# Tagged Questions

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

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### proving the output of cantor pairing function is un correlated if one bit of input is changed

Pairing function $\pi :N \times N\rightarrow N$ is defined as: $\pi(a_{1},a_{2})=\frac{1}{2}(a_{1}+a_{2})(a_{1}+a_{2}+1)+a_{2}$. Exp 1. If $a_{1},a_{2}$ are input then output is $\pi_{1}$ Exp 2. If ...
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### How to express $2x^3-x^2-3x+2$ as a linear combination of Legendre polynomials

I have used the formula \begin{align}p_0(x)&=1\\ p_1(x)&=x\\ p_2(x)&=\frac12(3x^2−1)\\ p_3(x)&=\frac12(5x^3−3x) \end{align} $$2x^3-x^2-3x+2=Ap_3(x)+Bp_2(x)+Cp_1(x)+Dp_0(x)$$ EDIT- \...
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### Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
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### Integral of incomplete gamma function and limit of hypergeometric function

Let $a > 0$ and consider the integral $$\int_x^\infty \frac{\Gamma(a,t)}{t}\,dt$$ where $\Gamma(a,t)$ is the upper incomplete gamma function $$\Gamma(a,t) = \int_t^\infty x^{a-1} e^{-x} \, dx.$$ ...
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### The Bessel function and finding expression

The Bessel Function $J_v$ of the first kind of order $v$ can be defined by the series expresion $$J_v(x)=\sum_{n=0} ^{\infty} \frac{(-1)^n}{n!\Gamma{(1+v+n)}}\left(\frac{x}{2}\right)^{2n+v}$$ (i) if ...
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### Integral ${\large\int}_0^\infty\big(2J_0(2x)^2+2Y_0(2x)^2-J_0(x)^2-Y_0(x)^2\big)\,dx$

I'm interested in the following definite integral: $$\int_0^\infty\big(2J_0(2x)^2-J_0(x)^2+2Y_0(2x)^2-Y_0(x)^2\big)\,dx,\tag1$$ where $J_\nu$ and $Y_\nu$ are the Bessel functions of the first and the ...
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### On series of the kind $\sum_{n=1}^\infty\frac{1}{n^2}\cdot\frac{1}{(1+nx)^s}$ and Frullani's theorem

I would like to know if my computations, in this post are not required justifications for the computations unless if there is a mistake in my claims, were rights and solve a question as if are known a ...
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### Upper bound for ratio of modified Bessel functions of second kind

I was wondering if someone has an idea if for $0 < x < y$, and $0< \nu \leq \frac{1}{2}$, one can obtain an upper bound for the ratio $$\frac{K_{\nu}(x)}{K_{\nu}(y)}$$ Thanks.
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### Solving $y^y = x$ for large $x$

I was playing around with recurrence relations and noticed that $\sqrt x$ has the fun property that $$\frac{x}{f(x)} = f(x)$$ ($\sqrt{x}$ and its negation are the only functions $f(x)$ that satisfy ...
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### An always increasing function

Suppose I wanted a function $f(x)$ such that the following properties are had. $f(x)$ maps $\mathbb{R}\to\mathbb{R}$. $f(a)>f(b)$ if $a>b$. The function may or may not be continuous, but it ...
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### Identity using q-Pochhammer symbols

Prove - $$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$ where $(a;q)$ are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ...
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### How to evaluate I(y) = $\int_0^t e^{ax^b} e^{-cx^d} x^f dx$ in terms of special functions?

To put the above in the proper context, I am trying to solve a Bernoulli equation of the second order: $\frac{dy}{dt} = -\frac{A}{p-q}(e^{-pt}-e^{-qt})y-Be^{-rt}y^2$ where constants A, B, p, q, r ...
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### How to compute the following integral?

Someone has an idea to calculate the following integral $$I_{a,b,\alpha} = \int_{1}^{+\infty} e^{-at} \,(1-t^{-1})^b \log^{\alpha}(1-t^{-1}) \, dt; \quad a,b>0, -1<\alpha<0.$$ Thank you in ...
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### How to compute the integral $I_{\alpha}$?

Someone has an idea to calculate the following integral $$I_{\alpha} = \int_{0}^{+\infty} t^{-\alpha} (1-a)^{t} dt; \quad 0<a,\alpha<1.$$ Thank you in advance
I came across the following ${}_3F_2$ hypergeometric polynomial: $${}_3F_2\left(\left.\begin{array}{c} 1,1,-n\\ 2, -1-2n \end{array}\right| -x\right)$$ for some large $x > 0$. I am wondering ...
Given the canonical infinite product representation (Weierstrass form) of the gamma function, \Gamma(z)= \left [ze^{\gamma z}\prod_{m=1}^{\infty} \left ( 1+ \frac{z}{m} \right)e^{-z/m} \right ]^{-1} ...