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

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0
votes
1answer
188 views

Is there an infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$ analogous to the Rogers-Ramanujan identity? [closed]

Given $$ \left(\frac{\eta(5\tau)}{\eta(\tau)}\right)^{6}\;\; =\;\; \frac{r^5}{1-11r^5-r^{10}},\;\;\;\;\;\text{with}\;\;r\; =\; q^{1/5} \prod_{n=1}^\infty ...
10
votes
4answers
653 views

Solving $(t^2+1)(y''-2y+1)=e^t$ with the initial conditions: $y(0)=y'(0)=1$

Since it is important to me I would like to award a user who would kindly explain me what are my mistakes and what is the correct way to solve the whole problem with 500 points. I'd really like your ...
8
votes
3answers
425 views

An infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$?

Given the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ where $q = \exp(2\pi i\tau)$. Consider the following "family", $\begin{align} ...
8
votes
3answers
484 views

Evaluation of $\sum_{n=1}^\infty \frac{1}{\Gamma (n+s)}$

I want to try and evaluate this interesting sum: $$\sum_{n=1}^\infty \frac{1}{\Gamma (n+s)}$$ where $0 \le s < 1$ WolframAlpha evaluates this sum to be $$\sum_{n=1}^\infty \frac{1}{\Gamma ...
0
votes
1answer
100 views

Finding a general coefficient in the multiplication of the two series

Help me please to find a general coefficient $a_j$ of the following series $$ ...
20
votes
1answer
531 views

Does $\left(n^2 \sin n\right)$ have a convergent subsequence?

I'm wrestling with the following: Question: For what values of $\alpha > 0$ does the sequence $\left(n^\alpha \sin n\right)$ have a convergent subsequence? (The special case $\alpha = 2$ in ...
6
votes
2answers
1k views

Are Complex Substitutions Legal in Integration?

This question has been irritating me for awhile so I thought I'd ask here. Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals: ...
4
votes
2answers
248 views

To find closed form of $\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt $

Let $x\geq 0$, then $$\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt = U_{\alpha} (x) $$ $$-\int_0^{\frac{\pi}{2}} \tan t \ e^{-x\tan t+\alpha t} \;dt = \frac{d (U_{\alpha} (x) )}{dx} $$ ...
7
votes
2answers
697 views

Approximate $\int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx $

I am trying to find an approximation to $$ I = \int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx. $$ My attempt is as follows: $$ \begin{align} I &= \int_a^b ...
2
votes
0answers
68 views

Is this formula for $ \sum_{n} (n^{2}+z^{2})^{-s} $ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
5
votes
2answers
157 views

Riemann zeta sums and harmonic numbers

Given the nth harmonic number of order s, $$H_n(s) =\sum_{m=1}^n \frac{1}{m^s}$$ It can be empirically observed that, for $s > 2$, then, $$\sum_{n=1}^\infty\Big[\zeta(s)-H_n(s)\Big] = ...
6
votes
1answer
296 views

On the Dirichlet beta function sum $\sum_{k=2}^\infty\Big[1-\beta(k) \Big]$

Given the Dirichlet beta function, $$\beta(k) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^k}$$ (The cases k = 2 is Catalan's constant.) It seems, $$\sum_{k=2}^\infty\Big[1-\beta(k) \Big] = ...
1
vote
0answers
85 views

integral with bessel function represented as a series [duplicate]

Possible Duplicate: prove equality with integral and series This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
2
votes
1answer
325 views

Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?

Motivation: I was looking at the approximation of the truncated Prime $\zeta$ function $$ P_x(s)=\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O \left(\cdot \right) $$ (to be found here with or ...
2
votes
1answer
116 views

To simplify $f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$

Let $x\leq0$, then $$ f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$$ $$ f'(x)= -\int_{-a}^{+a} \frac{1}{t^2-a^2} e^ {-\frac{x}{t^2-a^2}}dt$$ $$ f'(x)= -\int_{-a}^{+a} ...
1
vote
0answers
134 views

How to find a function with the following properties?

I want to find a function $f(s,x)$ such that $f(s,x)$ is analytic for any $s \in Z^+ $, $f(s,x)=B_s(x)$, where $B_s(x)$ are the Bernoulli polynomials $f(a, x)$ is elementary against $x$ at any ...
8
votes
2answers
2k views

Euler's product formula for $\sin(\pi z)$ and the gamma function

I want to derive Euler's infinite product formula $$\displaystyle \sin(\pi z) = \pi z \prod_{k=1}^\infty \left( 1 - \frac{z^2}{k^2} \right)$$ by using Euler's reflection equation ...
3
votes
0answers
207 views

Common zeros of associated Legendre functions

Suppose that $x_{0}$ is a zero of the associated Legendre function $P_{n}^{m}(x)$ (the degree $n$ is a positive integer while the order $m$ is an integer in the range from $0$ to $n$). If there exist ...
4
votes
1answer
143 views

Hypergeometric functions inequality

Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers. From a simple plot it looks like $_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} ...
5
votes
1answer
226 views

prove equality with integral and series

I am stuck on one question with integral. Help me please to show that with $n=1$ the following is true $$ ...
16
votes
4answers
507 views

The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$

Does anyone know if this function has a name? I came up with it by looking at the power series for $e^z$, changing the summation to an integral, and substituting the gamma function for the factorial ...
15
votes
1answer
367 views

On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others

For p = 2, we have, $\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$ It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then, ...
11
votes
1answer
223 views

A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation ...
4
votes
2answers
868 views

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ ...
0
votes
1answer
95 views

Condition for frame of $L_2$

Let $f$ be continuous, real valued and compactly supported with exactly one maximum function in $L_2$. Form the functions $$ f_{m,k}=f^m(x-2^k) $$ Under which conditions $\{f_{m,k}\}$ would be a ...
1
vote
0answers
706 views

solving Bessel function equation by hand

I have a Bessel function of the first kind given by the equation $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m+\alpha}$$ I am trying to ...
10
votes
3answers
322 views

Nicer expression for the following differential operator

I have the following sequence of differential operators: $$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$ Is there any expression involving a sum of "normal" ...
7
votes
0answers
160 views

Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

While working on mixture (variance) of normal distribution and keep running into these two integrals $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ ...
2
votes
2answers
390 views

Polynomials in Fourier trigonometric series

I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding: $\frac{\sin{kx}}{k}$, $\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$, $\frac{2 x ...
5
votes
2answers
229 views

What is the value of $\Gamma(\mathrm{i})$ ?

What is the value of $\Gamma(\mathrm{i})$ ? $\Gamma(z)$ is Gamma function. Here $\mathrm{i}^2=-1$.Can you help me with this problem ?
4
votes
1answer
117 views

Beta integral transformation

It's a homework task and I can't get past the last step. Task is to prove that $$ B(x,y)=\int\limits_0^1 \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau $$ By substituting ...
16
votes
3answers
734 views

An interesting sum to infinity

Is there any simple way of computing the following sum? $$\sum_{k=1}^\infty \frac1{k\space k!}$$
2
votes
0answers
74 views

Lower bound for the eigenvalue

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
4
votes
0answers
188 views

Weierstrass $\wp$-Function Addition Property

Consider the function $$ \det\left( \begin{array}{ccccc} &1 &\wp(z) &\wp'(z) \\ &1 &\wp(w) &\wp'(w) \\ &1 &\wp(-z-w) &\wp'(-z-w) \end{array} \right)=f(z) $$ I'm ...
2
votes
0answers
172 views

Hermite functions and integral

Let $$ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function. Consider $$ ...
3
votes
0answers
178 views

Equivalent Definitions of the Weierstass $\wp$-Function

I've come across two equivalent definitions of the Weierstrass $\wp$-function, but don't know how to prove that they are equivalent. Definition 1 $\wp(z)=cf(z)+d$ where $f$ is the elliptic function ...
1
vote
0answers
90 views

$n$-th derivative of the prolate spheroidal function

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
2
votes
2answers
200 views

Equality involving Appell hypergeometric function

After some algebra, Wolfram online integrator gave me the following: $$\tag{1} \int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t = c\ \cdot t^{3/2}\ \operatorname{F}_1 \left( \frac{3}{2}; -\frac{1}{2}, ...
0
votes
1answer
105 views

Expansions of Hermite functions

I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients. Thank you for your ...
4
votes
1answer
149 views

Perrin numbers in terms of the generalized hypergeometric function?

Given the roots of $x^3=x^2+1$, we have sequence A001609, $M(n) = x_1^n+x_2^n+x_3^n = \,_3F_2\left(\frac{-n}{3}, \frac{1-n}{3}, \frac{2-n}{3};\; \frac{1-n}{2}, \frac{2-n}{2};\; ...
6
votes
1answer
255 views

Integral with Bessel function

Let $n$ be half an odd integer, say $n=k+1/2, k \in \mathbb{N}$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral: $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot ...
5
votes
1answer
401 views

How to prove Gauss's Digamma Theorem?

Here $\psi(z)$ is digamma function, $\Gamma(z)$ is gamma function. $$\psi(z)=\frac{{\Gamma}'(z)}{\Gamma(z)},$$ For positive integers $m$ and $k$ (with $m < k$), the digamma function may be ...
3
votes
1answer
712 views

95% of energy of Bessel Functions

How can we determine what which Bessel function amplitudes contain the majority of the the energy? Similar to the Carson bandwidth rule, I want to determine which sidebands help make up the 95% of ...
5
votes
2answers
163 views

Procedure for evaluating the hypergeometric series $_2F_1\left\{\frac{v+2}{2},\frac{v+3}{2};v+1;z\right\}$

I'm trying to work out the procedure to get the following hypergeometric series into a simpler form, for all postive integer $v$: $$ _2F_1\left\{\frac{v+2}{2},\frac{v+3}{2};v+1;z\right\}$$ For ...
1
vote
0answers
297 views

Using Rouche's theorem

Let $p>1$. Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$. Function $\phi(p)$ is analytic on its domain. It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
2
votes
2answers
262 views

Relating Gamma and factorial function for non-integer values.

We have $$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$ for integers, so if $\Delta$ is some real value with $$0<\Delta<1,$$ then $$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$ because ...
0
votes
1answer
173 views

How to determinate the linearly independence between some special functions defined by ODE?

How to determinate the linearly independence between some special functions defined by ODE? For example: ${}_1F_1(a;b;x)$ , $x^{1-b}{}_1F_1(a-b+1;2-b;x)$ when $b$ is integer ${}_2F_1(a,b;c;x)$ , ...
2
votes
1answer
127 views

Is this series expressible in terms of Gauss' hypergeometric function?

How we can express this series $$F(z)=\sum_{n=0}^\infty \frac{z^n}{(a)_nn!}$$ in terms of Gauss' hypergeometric function? where $(a)_n$ denotes the Pochhammer symbol. Thanks in advance
2
votes
0answers
224 views

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference ...
4
votes
2answers
1k views

Why are the Gegenbauer polynomials called “ultraspherical”?

There has to be a good reason why the Gegenbauer polynomials were also named "ultraspherical" polynomials. I am aware that when $\alpha=\frac{1}{2}$, the Gegenbauer polynomials reduce to the Legendre ...