Laplace transform of generalized hypergeometric distribution
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What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below
$$p_X(x)=Kcdotfrac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}$$
for $theta>0$ and some constant $K$?
Meaning $$mathcal{L}left{ p_X(x)right} left(sright)=Kcdotsum_{xgeq0}frac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}e^{-xs}
$$ In the case you would know also the cumulant generating function $K(t)=log M(t)$, I would be most obliged.
laplace-transform hypergeometric-function moment-generating-functions cumulants
asked Nov 27 '18 at 18:18
Rafael Rafael
465
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add a comment |
$begingroup$
What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below
$$p_X(x)=Kcdotfrac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}$$
for $theta>0$ and some constant $K$?
Meaning $$mathcal{L}left{ p_X(x)right} left(sright)=Kcdotsum_{xgeq0}frac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}e^{-xs}
$$ In the case you would know also the cumulant generating function $K(t)=log M(t)$, I would be most obliged.
laplace-transform hypergeometric-function moment-generating-functions cumulants
asked Nov 27 '18 at 18:18
Rafael Rafael
465
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You need an integral sign just after the equal sign in the second formula...
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– Jean Marie
Nov 27 '18 at 18:36
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in the discrete case it is a sum e.g. here
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– Rafael
Nov 27 '18 at 18:40
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Your formulas are incomplete : you need a summation symbol Σ in your first equation and an integral sign $int_0^{infty}$... before summation symbol Σ in the second equation.
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– Jean Marie
Nov 27 '18 at 22:32
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my first equation is meant not to be a generalised hypergeometric function, but it terms. The constant is 1/$sum$. It is a function of x, not $theta$
$endgroup$
– Rafael
Nov 28 '18 at 9:34
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The terms are the probabilities.
$endgroup$
– Rafael
Nov 28 '18 at 9:35
add a comment |
$begingroup$
What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below
$$p_X(x)=Kcdotfrac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}$$
for $theta>0$ and some constant $K$?
Meaning $$mathcal{L}left{ p_X(x)right} left(sright)=Kcdotsum_{xgeq0}frac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}e^{-xs}
$$ In the case you would know also the cumulant generating function $K(t)=log M(t)$, I would be most obliged.
laplace-transform hypergeometric-function moment-generating-functions cumulants
asked Nov 27 '18 at 18:18
Rafael Rafael
465
$endgroup$
What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below
$$p_X(x)=Kcdotfrac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}$$
for $theta>0$ and some constant $K$?
Meaning $$mathcal{L}left{ p_X(x)right} left(sright)=Kcdotsum_{xgeq0}frac{(a_1)_xdots(a_p)_x}{(b_1)_xdots(b_q)_x}cdotfrac{theta^x}{x!}e^{-xs}
$$ In the case you would know also the cumulant generating function $K(t)=log M(t)$, I would be most obliged.
laplace-transform hypergeometric-function moment-generating-functions cumulants
laplace-transform hypergeometric-function moment-generating-functions cumulants
asked Nov 27 '18 at 18:18
Rafael Rafael
465
asked Nov 27 '18 at 18:18
Rafael Rafael
465
asked Nov 27 '18 at 18:18
Rafael Rafael
465
asked Nov 27 '18 at 18:18
Rafael Rafael
465
asked Nov 27 '18 at 18:18
Rafael Rafael
465
465
$begingroup$
You need an integral sign just after the equal sign in the second formula...
$endgroup$
– Jean Marie
Nov 27 '18 at 18:36
$begingroup$
in the discrete case it is a sum e.g. here
$endgroup$
– Rafael
Nov 27 '18 at 18:40
$begingroup$
Your formulas are incomplete : you need a summation symbol Σ in your first equation and an integral sign $int_0^{infty}$... before summation symbol Σ in the second equation.
$endgroup$
– Jean Marie
Nov 27 '18 at 22:32
$begingroup$
my first equation is meant not to be a generalised hypergeometric function, but it terms. The constant is 1/$sum$. It is a function of x, not $theta$
$endgroup$
– Rafael
Nov 28 '18 at 9:34
$begingroup$
The terms are the probabilities.
$endgroup$
– Rafael
Nov 28 '18 at 9:35
add a comment |
$begingroup$
You need an integral sign just after the equal sign in the second formula...
$endgroup$
– Jean Marie
Nov 27 '18 at 18:36
$begingroup$
in the discrete case it is a sum e.g. here
$endgroup$
– Rafael
Nov 27 '18 at 18:40
$begingroup$
Your formulas are incomplete : you need a summation symbol Σ in your first equation and an integral sign $int_0^{infty}$... before summation symbol Σ in the second equation.
$endgroup$
– Jean Marie
Nov 27 '18 at 22:32
$begingroup$
my first equation is meant not to be a generalised hypergeometric function, but it terms. The constant is 1/$sum$. It is a function of x, not $theta$
$endgroup$
– Rafael
Nov 28 '18 at 9:34
$begingroup$
The terms are the probabilities.
$endgroup$
– Rafael
Nov 28 '18 at 9:35
$begingroup$
You need an integral sign just after the equal sign in the second formula...
$endgroup$
– Jean Marie
Nov 27 '18 at 18:36
$begingroup$
You need an integral sign just after the equal sign in the second formula...
$endgroup$
– Jean Marie
Nov 27 '18 at 18:36
$begingroup$
in the discrete case it is a sum e.g. here
$endgroup$
– Rafael
Nov 27 '18 at 18:40
$begingroup$
in the discrete case it is a sum e.g. here
$endgroup$
– Rafael
Nov 27 '18 at 18:40
$begingroup$
Your formulas are incomplete : you need a summation symbol Σ in your first equation and an integral sign $int_0^{infty}$... before summation symbol Σ in the second equation.
$endgroup$
– Jean Marie
Nov 27 '18 at 22:32
$begingroup$
Your formulas are incomplete : you need a summation symbol Σ in your first equation and an integral sign $int_0^{infty}$... before summation symbol Σ in the second equation.
$endgroup$
– Jean Marie
Nov 27 '18 at 22:32
$begingroup$
my first equation is meant not to be a generalised hypergeometric function, but it terms. The constant is 1/$sum$. It is a function of x, not $theta$
$endgroup$
– Rafael
Nov 28 '18 at 9:34
$begingroup$
my first equation is meant not to be a generalised hypergeometric function, but it terms. The constant is 1/$sum$. It is a function of x, not $theta$
$endgroup$
– Rafael
Nov 28 '18 at 9:34
$begingroup$
The terms are the probabilities.
$endgroup$
– Rafael
Nov 28 '18 at 9:35
$begingroup$
The terms are the probabilities.
$endgroup$
– Rafael
Nov 28 '18 at 9:35
add a comment |
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$begingroup$
You need an integral sign just after the equal sign in the second formula...
$endgroup$
– Jean Marie
Nov 27 '18 at 18:36
$begingroup$
in the discrete case it is a sum e.g. here
$endgroup$
– Rafael
Nov 27 '18 at 18:40
$begingroup$
Your formulas are incomplete : you need a summation symbol Σ in your first equation and an integral sign $int_0^{infty}$... before summation symbol Σ in the second equation.
$endgroup$
– Jean Marie
Nov 27 '18 at 22:32
$begingroup$
my first equation is meant not to be a generalised hypergeometric function, but it terms. The constant is 1/$sum$. It is a function of x, not $theta$
$endgroup$
– Rafael
Nov 28 '18 at 9:34
$begingroup$
The terms are the probabilities.
$endgroup$
– Rafael
Nov 28 '18 at 9:35