The Laplace Transform(s) of a Certain Family of Generalized Hypergeometric Functions
$begingroup$
Using the standard notation for a generalized hypergeometric function, given a non-negative integer $p$, define:
$mathcal{G}_{p}left(xright)={}_{p}F_{p}left(underbrace{frac{1}{2},...,frac{1}{2}}_{ptextrm{ times}};underbrace{frac{3}{2},...,frac{3}{2}}_{ptextrm{ times}};-x^{2}right)$
for all $x$ (in $mathbb{R}$, or in $mathbb{C}$). That is to say (as can be easily shown):
$mathcal{G}_{p}left(xright)=3^{p}sum_{n=0}^{infty}frac{left(-1right)^{n}}{n!}frac{x^{2n}}{left(2n+3right)^{p}}$
This function is integrable in the nicest possible ways (it is a schwartz function, rapidly decreasing, &c., &c....) and, it is analytic everywhere.
I would like to be able to find the laplace transforms ($mathcal{L}left{ mathcal{G}_{p}right} left(sright)
)$ of the $mathcal{G}_{p}$s for every $p$
. At a minimum, I want a closed-form expression for the value of the laplace transforms at $s=1$
; i.e., the value of the integral:
$int_{0}^{infty}mathcal{G}_{p}left(xright)e^{-x}dx$
I cannot do this integration term-by-term, since that leads to a divergent series (and is not valid, due to an absence of uniform convergence of the integrand for $xinmathbb{R}geq0$).
Any ideas?
laplace-transform hypergeometric-function
asked Mar 6 '17 at 1:43
MCSMCS
969313
$endgroup$
add a comment |
$begingroup$
Using the standard notation for a generalized hypergeometric function, given a non-negative integer $p$, define:
$mathcal{G}_{p}left(xright)={}_{p}F_{p}left(underbrace{frac{1}{2},...,frac{1}{2}}_{ptextrm{ times}};underbrace{frac{3}{2},...,frac{3}{2}}_{ptextrm{ times}};-x^{2}right)$
for all $x$ (in $mathbb{R}$, or in $mathbb{C}$). That is to say (as can be easily shown):
$mathcal{G}_{p}left(xright)=3^{p}sum_{n=0}^{infty}frac{left(-1right)^{n}}{n!}frac{x^{2n}}{left(2n+3right)^{p}}$
This function is integrable in the nicest possible ways (it is a schwartz function, rapidly decreasing, &c., &c....) and, it is analytic everywhere.
I would like to be able to find the laplace transforms ($mathcal{L}left{ mathcal{G}_{p}right} left(sright)
)$ of the $mathcal{G}_{p}$s for every $p$
. At a minimum, I want a closed-form expression for the value of the laplace transforms at $s=1$
; i.e., the value of the integral:
$int_{0}^{infty}mathcal{G}_{p}left(xright)e^{-x}dx$
I cannot do this integration term-by-term, since that leads to a divergent series (and is not valid, due to an absence of uniform convergence of the integrand for $xinmathbb{R}geq0$).
Any ideas?
laplace-transform hypergeometric-function
asked Mar 6 '17 at 1:43
MCSMCS
969313
$endgroup$
add a comment |
$begingroup$
Using the standard notation for a generalized hypergeometric function, given a non-negative integer $p$, define:
$mathcal{G}_{p}left(xright)={}_{p}F_{p}left(underbrace{frac{1}{2},...,frac{1}{2}}_{ptextrm{ times}};underbrace{frac{3}{2},...,frac{3}{2}}_{ptextrm{ times}};-x^{2}right)$
for all $x$ (in $mathbb{R}$, or in $mathbb{C}$). That is to say (as can be easily shown):
$mathcal{G}_{p}left(xright)=3^{p}sum_{n=0}^{infty}frac{left(-1right)^{n}}{n!}frac{x^{2n}}{left(2n+3right)^{p}}$
This function is integrable in the nicest possible ways (it is a schwartz function, rapidly decreasing, &c., &c....) and, it is analytic everywhere.
I would like to be able to find the laplace transforms ($mathcal{L}left{ mathcal{G}_{p}right} left(sright)
)$ of the $mathcal{G}_{p}$s for every $p$
. At a minimum, I want a closed-form expression for the value of the laplace transforms at $s=1$
; i.e., the value of the integral:
$int_{0}^{infty}mathcal{G}_{p}left(xright)e^{-x}dx$
I cannot do this integration term-by-term, since that leads to a divergent series (and is not valid, due to an absence of uniform convergence of the integrand for $xinmathbb{R}geq0$).
Any ideas?
laplace-transform hypergeometric-function
asked Mar 6 '17 at 1:43
MCSMCS
969313
$endgroup$
Using the standard notation for a generalized hypergeometric function, given a non-negative integer $p$, define:
$mathcal{G}_{p}left(xright)={}_{p}F_{p}left(underbrace{frac{1}{2},...,frac{1}{2}}_{ptextrm{ times}};underbrace{frac{3}{2},...,frac{3}{2}}_{ptextrm{ times}};-x^{2}right)$
for all $x$ (in $mathbb{R}$, or in $mathbb{C}$). That is to say (as can be easily shown):
$mathcal{G}_{p}left(xright)=3^{p}sum_{n=0}^{infty}frac{left(-1right)^{n}}{n!}frac{x^{2n}}{left(2n+3right)^{p}}$
This function is integrable in the nicest possible ways (it is a schwartz function, rapidly decreasing, &c., &c....) and, it is analytic everywhere.
I would like to be able to find the laplace transforms ($mathcal{L}left{ mathcal{G}_{p}right} left(sright)
)$ of the $mathcal{G}_{p}$s for every $p$
. At a minimum, I want a closed-form expression for the value of the laplace transforms at $s=1$
; i.e., the value of the integral:
$int_{0}^{infty}mathcal{G}_{p}left(xright)e^{-x}dx$
I cannot do this integration term-by-term, since that leads to a divergent series (and is not valid, due to an absence of uniform convergence of the integrand for $xinmathbb{R}geq0$).
Any ideas?
laplace-transform hypergeometric-function
laplace-transform hypergeometric-function
asked Mar 6 '17 at 1:43
MCSMCS
969313
asked Mar 6 '17 at 1:43
MCSMCS
969313
asked Mar 6 '17 at 1:43
MCSMCS
969313
asked Mar 6 '17 at 1:43
MCSMCS
969313
asked Mar 6 '17 at 1:43
MCSMCS
969313
969313
add a comment |
add a comment |
1 Answer
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$begingroup$
Following Wang, I would suggest
$$mathcal{L}{mathcal{G}_{p}(x);s}= _{p+1}F_{q}left[begin{matrix}
1, & a_{1}, & dots & ,a_{n} \
& b_{1}, & dots & ,b_{q}
end{matrix};1/sright]$$
answered Nov 27 '18 at 18:00
Rafael Rafael
465
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add a comment |
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1 Answer
1
active
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votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Following Wang, I would suggest
$$mathcal{L}{mathcal{G}_{p}(x);s}= _{p+1}F_{q}left[begin{matrix}
1, & a_{1}, & dots & ,a_{n} \
& b_{1}, & dots & ,b_{q}
end{matrix};1/sright]$$
answered Nov 27 '18 at 18:00
Rafael Rafael
465
$endgroup$
add a comment |
$begingroup$
Following Wang, I would suggest
$$mathcal{L}{mathcal{G}_{p}(x);s}= _{p+1}F_{q}left[begin{matrix}
1, & a_{1}, & dots & ,a_{n} \
& b_{1}, & dots & ,b_{q}
end{matrix};1/sright]$$
answered Nov 27 '18 at 18:00
Rafael Rafael
465
$endgroup$
add a comment |
$begingroup$
Following Wang, I would suggest
$$mathcal{L}{mathcal{G}_{p}(x);s}= _{p+1}F_{q}left[begin{matrix}
1, & a_{1}, & dots & ,a_{n} \
& b_{1}, & dots & ,b_{q}
end{matrix};1/sright]$$
answered Nov 27 '18 at 18:00
Rafael Rafael
465
$endgroup$
Following Wang, I would suggest
$$mathcal{L}{mathcal{G}_{p}(x);s}= _{p+1}F_{q}left[begin{matrix}
1, & a_{1}, & dots & ,a_{n} \
& b_{1}, & dots & ,b_{q}
end{matrix};1/sright]$$
answered Nov 27 '18 at 18:00
Rafael Rafael
465
answered Nov 27 '18 at 18:00
Rafael Rafael
465
answered Nov 27 '18 at 18:00
Rafael Rafael
465
answered Nov 27 '18 at 18:00
Rafael Rafael
465
465
add a comment |
add a comment |
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