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Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
2
votes
2
answers
204
views
Zeroes of trigonometric-like function
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 …
5
votes
1
answer
295
views
Estimate of incomplete binomial integral
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}{ …
3
votes
Inverse Laplace transform of a hypergeometric function
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 …
0
votes
Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$
There is a proof using only change of variables in N.N.Lebedev's book (p.28 of the Russian edition).
5
votes
2
answers
607
views
Integrals involving trigonometric functions and polynomials
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 …
5
votes
Evaluating elliptic integrals
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 …
7
votes
Accepted
A (likely) positivity property of the Lerch zeta-function
From Prudnikov,Brychkov,Marichev Integral and Series, Vol.1, sect. 5.4.3, formula 1 we derive that
$$
\sum_{k=0}^\infty \frac{\cos(ak)}{(k+1/2)^{p+1}}=
\frac{1}{\Gamma(s)}\int_0^{\infty}\frac{t^p e^{- …
1
vote
Is there any simpler form of this function
First apply the Pfaff transformation mentioned above:
$$
F(2n)=1/2\ _2F_1(1,n+1;\frac{(n+1)+n}{2};1/2).
$$
Then use a formula from section 7.3.8 in Prudnikov,Bychkov,Marichev Integral and series, vol …
2
votes
What function is "$U_{\nu}(\cdot, \cdot)$"?
This is the Lommel function of two variables, cf. p.748 of the book you mentioned for its definition.
1
vote
Integral involving exponential and Marcum-Q function
Making a change in the sum $l+k=j$ we immediately evaluate this integral in terms of the Appell hypergeometric function, if the aim was to classify it via something known:
$$
I=\frac{1}{2}\sqrt{\frac{ …
0
votes
On a polynomial related to the Legendre function of the second kind
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 …
3
votes
Accepted
Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients
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 …
0
votes
About Turan`s problem(inequality) in multivariable
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( …
2
votes
Accepted
Estimate of a ratio of two incomplete gamma functions
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)}{ …
2
votes
Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$
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 …