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Results tagged with special-functions
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user 49208
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.
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 …
- 1,560
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 …
- 1,560
9
votes
Recognize this sum
This is so-called generalized Mittag-Leffler function, more exactly the Wright function (as series ) or the Fox function (as inverse Mellin transform). A lot is known about them
Start with
1. http:// …
- 1,560
2
votes
2
answers
240
views
Evaluate an integral or Fourier coefficients
Consider an integral
$$
\int_0^\pi \frac{\cos(kx)}{\cosh(ax)}\ dx
$$
there $k\in
\mathbb{Z}, a\in \mathbb{R}.$
Of course that is Fourier coefficient for the function $f(x)=\frac{1}{\cosh(ax)}.$
Ques …
- 1,560
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 …
- 1,560
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:
$$
I=\frac{1}{2}\sqrt{\frac{ …
- 1,560
3
votes
Logarithm of the hypergeometric function
Please note the paper, may be it will be useful:
D. Karp, S.M. Sitnik,
Log-convexity and log-concavity of hypergeometric-like functions,
Journal of Mathematical Analysis and Applications, Volume 364, …
- 1,560
3
votes
Inverse Laplace transform of a hypergeometric function
Using known reference books we derive that in this case $_2F_1(a,1-a;1+a;z)$ is reduced to incomplete Beta-function $B_{\frac{1-z}{2}}(a,a)$. That is much simpler and easy to estimate exactly or numer …
- 1,560
2
votes
Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$
In this case series are not convergent. And the Gauss function is not defined by series outside the unit circle. But there are explicit analytical continuation formulas to return inside of the unit ci …
- 1,560
0
votes
Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$
There is a proof using only change of variables in N.N.Lebedev's book (p.28 of the Russian edition).
- 1,560
2
votes
Accepted
Estimate of a ratio of two incomplete gamma functions
In fact it seems to be a consequence of the Cauchy theorem from calculus. Really, by it and a formula for derivative of incomplete gamma function (cf. Wiki for example) we evaluate
$\frac{f(x)-f(y)}{ …
- 1,560
0
votes
About Turan`s problem(inequality) in multivariable
For the case of a single variable an obvious condition on polynomials is that a ratio
$$
\frac{P_{n-1}(x)P_{n+1}(x)}{(P_n(x))^2}
$$
is monotone. Then sharp estimates hold true with limits via $P_n( …
- 1,560
2
votes
What function is "$U_{\nu}(\cdot, \cdot)$"?
This is the Lommel function of two variables, cf. p.748 of the book you mentioned for its definition.
- 1,560
0
votes
A definite integral of hypergeometric function 2F1
May be use the series for $_{2}F_{1}$ and then integrate term by term? The result looks like a sort of Appell-type function of two arguments: $x, 1-x$...
- 1,560
1
vote
Conjectured bound on Kummer's function (confluent hypergeometric function)
1) Cf. inequalities in the paper:
D. Karp, S.M. Sitnik, Log-convexity and log-concavity of hypergeometric-like functions, Journal of Mathematical Analysis and Applications, Volume 364, Issue 2, P. 384 …
- 1,560