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Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT A …

Let $0\le k \le n$. Prove that $$ n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2. $$ As far as I know 1) it is proved for $\frac{k}{n+1}\le 1/2$ and 2) not proved for $1/2 <\frac{k}{ …

Using known reference books we derive that in this case $_2F_1(a,1-a;1+a;z)$ is reduced to incomplete Beta-function $B_{\frac{1-z}{2}}(a,a)$. That is much simpler and easy to estimate exactly or numer …

There is a proof using only change of variables in N.N.Lebedev's book (p.28 of the Russian edition).

It seems to be known as symmetric elliptic integrals of Carlson. Look in the NIST book, 19.15 and further. There are a lot of formulas in it. It seems you seek for exactly the formula 19.22.8 on page …

From Prudnikov,Brychkov,Marichev Integral and Series, Vol.1, sect. 5.4.3, formula 1 we derive that $$ \sum_{k=0}^\infty \frac{\cos(ak)}{(k+1/2)^{p+1}}= \frac{1}{\Gamma(s)}\int_0^{\infty}\frac{t^p e^{- …

First apply the Pfaff transformation mentioned above: $$ F(2n)=1/2\ _2F_1(1,n+1;\frac{(n+1)+n}{2};1/2). $$ Then use a formula from section 7.3.8 in Prudnikov,Bychkov,Marichev Integral and series, vol …

Can one describe all the real polynomials $P(x)$ such that the following integrals converge: $$ \int_0^{\infty} \sin(P(x))dx, \int_0^{\infty} \cos(P(x))dx ? $$ Among special cases are such celebriti …

This is the Lommel function of two variables, cf. p.748 of the book you mentioned for its definition.

Making a change in the sum $l+k=j$ we immediately evaluate this integral in terms of the Appell hypergeometric function, if the aim was to classify it via something known: $$ I=\frac{1}{2}\sqrt{\frac{ …

The explicit formular for $W_n(x)$ is known, it is on p. 360 of the NIST Handbook of Mathematical Functions, formulas (14.7.2)--(14.7.7). It is in fact an explicit polynomial with coefficients depend …

The formular you search for is really known. It is on p. 360 of the NIST Handbook of Mathematical Functions, formulas (14.7.2)--(14.7.7). $p_n(x)$ is in fact an explicit polynomial (not rational) wit …

For the case of a single variable an obvious condition on polynomials is that a ratio $$ \frac{P_{n-1}(x)P_{n+1}(x)}{(P_n(x))^2} $$ is monotone. Then sharp estimates hold true with limits via $P_n( …

In fact it seems to be a consequence of the Cauchy theorem from calculus. Really, by it and a formula for derivative of incomplete gamma function (cf. Wiki for example) we evaluate $\frac{f(x)-f(y)}{ …

In this case series are not convergent. And the Gauss function is not defined by series outside the unit circle. But there are explicit analytical continuation formulas to return inside of the unit ci …