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The existing literature on generalized trigonometric functions is scarce and it seems there isn't a comprehensive account of generalized trigonometric functions anywhere. Also there isn't a unified ac …

By periodicity of trigonometric functions we have $$ I(m,n)=\int\limits_0^{2\pi}P_m(\cos{\alpha})\,\cos^{m+2}{(n\alpha+n\theta)}\;d\alpha.\tag{1} $$ Moreover, by the symmetry of Legendre polynomials: …

This integral is due to Lobachevskii. He gave it in more general form as follows which can be found in his work "Application of imaginary geometry to certain integrals" (1836). Also see equation 3.84 …

This is particular case of a classic integral studied by Ramanujan. See Chapter 11 in Hardy's book, "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work", where it is shown that $$ \ …

Start from the well known formula \begin{equation} 2^{N-1} \prod _{k=1}^N \left[\cos (x-y)-\cos \left(x+y+\frac{2\pi k}{N}\right)\right]=\cos N(x-y)-\cos N(x+y), \end{equation} take logarithmic de …

The only odd values of $\phi(n)$ are $\phi(1)=\phi(2)=1$. $\phi(n)$ is even but not divisible by $4$ when: $n=4$ $n=2^{\left\{0,1\right\}}p^m$, where $p=4k+3$ is prime, $m=1,2,3,...$ We have $$ \ …

A more general result is due to C. Störmer (Acta Mathematica December 1895, Volume 19, Issue 1, pp 341–350)

The generalization looks like this $$ \int_{-\infty}^{\infty} \binom{n}{\alpha x}^l dx =\sum_{k=-\infty}^\infty\binom{n}{\alpha k}^l,\quad 0<\alpha\le 2/l,~l\in\mathbb{N}\tag{1} $$ where $n$ need not …

There is a similar formula $$ \small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}^{n-m}\frac{(-1)^k}{4^{k+m}}\binom{n+m}{k+2m}\binom{n+k+m}{n+m}\binom{2k+2m}{k+ …

Since the main contribution to the integral comes from $t<<1$, analytically one has \begin{align} \lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt&=\lim_{c\to 0^+}\int_c^{\pi/2}\fr …

This problem can be reduced at least formally to a compact double integral, which might be easier to solve. Starting with the integral representation for the Gamma function, we write the double sum a …

This question has been completely reformulated and a new property for the function $f_q$ has been added due to a series of helpful comments by fedja. Consider the integral from quantum field theory …

The function $$ f_q(x,y,z)=\sum_{cyc}e^z\frac{\theta_q\left(e^{\frac{2 \pi i}{3}+x-z}\right) \theta_q\left(e^{\frac{2 \pi i}{3}+y-z}\right)}{\theta_q\left(e^{x-z}\right) \theta_q\left(e^{y-z}\right) …

In the paper Transformations of infinite series, Bryden Cais gives the following transformations of infinite products Theorem 4. If $$ f(t) = \frac{\cosh(\pi t)-1}{\sinh(\pi t)}\frac{\cosh(2\pi t)+1} …

EDIT (Feb 2024): The question has been generally answered in the article: Martin Nicholson, Finite and infinite product transformations, arXiv:1712.06097. Question $2$ has a surprisingly simple ans …