4 H-Functions

4.1 Definition

[120.1.1] The $H$ -function of order $(m,n,p,q)\in\mathbb{N}^{4}$ and with parameters $A_{i}\in\mathbb{R}_{+}(i=1,\ldots,p)$ , $B_{i}\in\mathbb{R}_{+}(i=1,\ldots,q)$ , $a_{i}\in\mathbb{C}(i=1,\ldots,p)$ , and $b_{i}\in\mathbb{C}(i=1,\ldots,q)$ is defined for $z\in\mathbb{C},z\neq 0$ by the contour integral [55, 56, 57, 58, 59]

$H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=\frac{1}{2\pi i}\int _{{\mathcal{L}}}\eta(s)z^{{-s}}\;\mbox{\rm d}s$

(153)

where the integrand is

$\eta(s)=\frac{\displaystyle\prod _{{i=1}}^{m}\Gamma(b_{i}+B_{i}s)\prod _{{i=1}}^{n}\Gamma(1-a_{i}-A_{i}s)}{\displaystyle\prod _{{i=n+1}}^{p}\Gamma(a_{i}+A_{i}s)\prod _{{i=m+1}}^{q}\Gamma(1-b_{i}-B_{i}s)}.$

(154)

[page 121, §0] [121.0.1] In (153) $z^{{-s}}=\exp\{-s\log|z|-i\arg z\}$ and $\arg z$ is not necessarily the principal value. [121.0.2] The integers $m,n,p,q$ must satisfy

$0\leq m\leq q,\qquad 0\leq n\leq p,$

(155)

and empty products are interpreted as being unity. [121.0.3] The parameters are restricted by the condition

$\mathbb{P}_{a}\cap\mathbb{P}_{b}=\emptyset$

(156)

where

$\displaystyle\mathbb{P}_{a}$	$\displaystyle=\{\text{poles of }\Gamma(1-a_{i}-A_{i}s)\}=\left\{\frac{1-a_{i}+k}{A_{i}}\in\mathbb{C}:i=1,\ldots,n;k\in\mathbb{N}_{0}\right\}$
$\displaystyle\mathbb{P}_{b}$	$\displaystyle=\{\text{poles of }\Gamma(b_{i}+B_{i}s)\}=\left\{\frac{-b_{i}-k}{B_{i}}\in\mathbb{C}:i=1,\ldots,m;k\in\mathbb{N}_{0}\right\}$		(157)

are the poles of the numerator in (154). [121.0.4] The integral converges if one of the following conditions holds [59]

$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(c-i\infty,c+i\infty;\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }\|\arg z\|<C\pi/2;\text{\ \ \ \ }C>0$		(158a)
$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(c-i\infty,c+i\infty;\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }\|\arg z\|=C\pi/2;\text{\ \ \ \ }C\geq 0;\text{\ \ \ \ }cD<-\mathrm{Re}\, F$		(158b)

$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(-\infty+i\gamma _{1},-\infty+i\gamma _{2};\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }D>0;\text{\ \ \ \ }0<\|z\|<\infty$	(159a)
$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(-\infty+i\gamma _{1},-\infty+i\gamma _{2};\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }D=0;\text{\ \ \ \ }0<\|z\|<E^{{-1}}$	(159b)
$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(-\infty+i\gamma _{1},-\infty+i\gamma _{2};\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }D=0;\text{\ \ \ \ }\|z\|=E^{{-1}};\text{\ }C\geq 0;\text{\ }\mathrm{Re}\, F<0$	(159c)

$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(\infty+i\gamma _{1},\infty+i\gamma _{2};\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }D<0;\text{\ \ \ \ }0<\|z\|<\infty$	(160a)
$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(\infty+i\gamma _{1},\infty+i\gamma _{2};\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }D=0;\text{\ \ \ \ }\|z\|>E^{{-1}}$	(160b)
$\displaystyle{\mathcal{L}}$	$\displaystyle={\mathcal{L}}(\infty+i\gamma _{1},\infty+i\gamma _{2};\mathbb{P}_{a},\mathbb{P}_{b});\text{\ \ \ \ }D=0;\text{\ \ \ \ }\|z\|=E^{{-1}};\text{\ }C\geq 0;\text{\ }\mathrm{Re}\, F<0$	(160c)

where $\gamma _{1}<\gamma _{2}$ . [121.0.5] Here ${\mathcal{L}}(z_{1},z_{2};\mathbb{G}_{1},\mathbb{G}_{2})$ denotes a contour in the complex plane starting at $z_{1}$ and ending at $z_{2}$ and separating the points in $\mathbb{G}_{1}$ from those

[page 122, §0] in $\mathbb{G}_{2}$ , and the notation

$\displaystyle C$	$\displaystyle=\sum _{{i=1}}^{n}A_{i}-\sum _{{i=n+1}}^{p}A_{i}+\sum _{{i=1}}^{m}B_{i}-\sum _{{i=m+1}}^{q}B_{i}$	(161)
$\displaystyle D$	$\displaystyle=\sum _{{i=1}}^{q}B_{i}-\sum _{{i=1}}^{p}A_{i}$	(162)
$\displaystyle E$	$\displaystyle=\prod _{{i=1}}^{p}A^{{A_{i}}}_{i}\prod _{{i=1}}^{q}B^{{-B_{i}}}_{i}$	(163)
$\displaystyle F$	$\displaystyle=\sum _{{i=1}}^{q}b_{i}-\sum _{{i=1}}^{p}a_{j}+(p-q)/2+1$	(164)

was employed. [122.0.1] The $H$ -functions are analytic for $z\neq 0$ and multivalued (single valued on the Riemann surface of $\log z$ ).

4.2 Basic Properties

[122.1.1] From the definition of the $H$ -functions follow some basic properties. [122.1.2] Let $S_{n}(n\geq 1)$ denote the symmetric group of $n$ elements, and let $\pi _{n}$ denote a permutation in $S_{n}$ . [122.1.3] Then the product structure of (154) implies that for all $\pi _{n}\in S_{n},\pi _{m}\in S_{m},\pi _{{p-n}}\in S_{{p-n}}$ and $\pi _{{q-m}}\in S_{{q-m}}$

$H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{P_{n},P_{{p-n}}}\\ {P_{m},P_{{q-m}}}\end{array}\right.\right)$

(165)

where the parameter permutations

$\displaystyle P_{n}$	$\displaystyle=(a_{{\pi _{n}(1)}},A_{{\pi _{n}(1)}}),\ldots,(a_{{\pi _{n}(n)}},A_{{\pi _{n}(n)}})$
$\displaystyle P_{{p-n}}$	$\displaystyle=(a_{{\pi _{{p-n}}(n+1)}},A_{{\pi _{{p-n}}(n+1)}}),\ldots,(a_{{\pi _{{p-n}}(p)}},A_{{\pi _{{p-n}}(p)}})$
$\displaystyle P_{m}$	$\displaystyle=(b_{{\pi _{m}(1)}},B_{{\pi _{m}(1)}}),\ldots,(b_{{\pi _{m}(m)}},B_{{\pi _{m}(m)}})$	(166)
$\displaystyle P_{{q-m}}$	$\displaystyle=(b_{{\pi _{{q-m}}(m+1)}},B_{{\pi _{{q-m}}(m+1)}}),\ldots,(b_{{\pi _{{q-m}}(q)}},B_{{\pi _{{q-m}}(q)}})$

have to be inserted on the right hand side. [122.1.4] If any of $n,m,p-n$ or $q-m$ vanishes the corresponding permutation is absent.

[122.2.1] The order reduction formula

$H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),(a_{2},A_{2})\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),(b_{2},B_{2})\ldots,(b_{{q-1}},B_{{q-1}})(a_{1},A_{1})}\end{array}\right.\right)=H^{{m,n-1}}_{{p-1,q-1}}\left(z\left|\begin{array}[]{l}{(a_{2},A_{2}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{{q-1}},B_{{q-1}})}\end{array}\right.\right)$

(167)

[page 123, §0] holds for $n\geq 1$ and $q>m$ , and similarly

$H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),(a_{2},A_{2})\ldots,(a_{{p-1}},A_{{p-1}})(b_{1},B_{1})}\\ {(b_{1},B_{1}),(b_{2},B_{2})\ldots,(b_{q},B_{q})}\end{array}\right.\right)=H^{{m-1,n}}_{{p-1,q-1}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{{p-1}},A_{{p-1}})}\\ {(b_{2},B_{2}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)$

(168)

for $m\geq 1$ and $p>n$ . [123.0.1] The formula

$H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a,0),(a_{2},A_{2})\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=\Gamma(1-a)H^{{m,n-1}}_{{p-1,q}}\left(z\left|\begin{array}[]{l}{(a_{2},A_{2}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)$

(169)

holds for $n\geq 1$ and $\mathrm{Re}\,(1-a)>0$ . [123.0.2] Analogous formulae are readily found if a parameter pair $(a,0)$ or $(b,0)$ appears in one of the other groups.

[123.1.1] A change of variables in (153) shows

$H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=H^{{n,m}}_{{q,p}}\left(\frac{1}{z}\left|\begin{array}[]{l}{(1-b_{1},B_{1}),\ldots,(1-b_{q},B_{q})}\\ {(1-a_{1},A_{1}),\ldots,(1-a_{p},A_{p})}\end{array}\right.\right)$

(170)

which allows to transform an $H$ -function with $D>0$ and $\arg z$ to one with $D<0$ and $\arg(1/z)$ . [123.1.2] For $\gamma>0$

$\frac{1}{\gamma}H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=H^{{m,n}}_{{p,q}}\left(z^{\gamma}\left|\begin{array}[]{l}{(a_{1},\gamma A_{1}),\ldots,(a_{p},\gamma A_{p})}\\ {(b_{1},\gamma B_{1}),\ldots,(b_{q},\gamma B_{q})}\end{array}\right.\right)$

(171)

while for $\gamma\in\mathbb{R}$

$z^{\gamma}H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=H^{{m,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1}+\gamma A_{1},A_{1}),\ldots,(a_{p}+\gamma A_{p},A_{p})}\\ {(b_{1}+\gamma B_{1},B_{1}),\ldots,(b_{q}+\gamma B_{q},B_{q})}\end{array}\right.\right)$

(172)

[page 124, §0] holds.

[124.1.1] For $m=0$ with conditions (159) the integrand is analytic and thus

$H^{{0,n}}_{{p,q}}\left(z\left|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=0.$

(173)

4.3 Integral Transformations

[124.2.1] The definition of an $H$ -function in eq. (153) becomes an inverse Mellin transform if ${\mathcal{L}}$ is chosen parallel to the imaginary axis inside the strip

$\max _{{1\leq i\leq m}}\mathrm{Re}\,\frac{-b_{i}}{B_{i}}<s<\min _{{1\leq i\leq m}}\mathrm{Re}\,\frac{1-a_{i}}{A_{i}}$

(174)

by the Mellin inversion theorem [60]. [124.2.2] Therefore

${\mathcal{M}}\left\{ H^{{m,n}}_{{p,q}}(z)\right\}(s)=\eta(s)=\frac{\displaystyle\prod _{{i=1}}^{m}\Gamma(b_{i}+B_{i}s)\prod _{{i=1}}^{n}\Gamma(1-a_{i}-A_{i}s)}{\displaystyle\prod _{{i=n+1}}^{p}\Gamma(a_{i}+A_{i}s)\prod _{{i=m+1}}^{q}\Gamma(1-b_{i}-B_{i}s)}$

(175)

whenever the inequality

$\max _{{1\leq i\leq m}}\mathrm{Re}\,\frac{-b_{i}}{B_{i}}<\min _{{1\leq i\leq m}}\mathrm{Re}\,\frac{1-a_{i}}{A_{i}}$

(176)

is fulfilled.

[124.3.1] The Laplace transform of an $H$ -function is obtained from eq. (175) by using eq. (78). [124.3.2] One finds

	$\displaystyle{\mathcal{L}}\left\{ H^{{m,n}}_{{p,q}}(z)\right\}(u)=\int _{0}^{\infty}e^{{-ux}}H^{{m,n}}_{{p,q}}\left(x\left\|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)\mbox{\rm d}x$
	$\displaystyle=H^{{n+1,m}}_{{q,p+1}}\left(u\left\|\begin{array}[]{l}{(1-b_{1}-B_{1},B_{1}),\ldots,(1-b_{q}-B_{q},B_{q})}\\ {(0,1)(1-a_{1}-A_{1},A_{1}),\ldots,(1-a_{p}-A_{p},A_{p})}\end{array}\right.\right)$
	$\displaystyle=\frac{1}{u}H^{{m,n+1}}_{{p+1,q}}\left(\frac{1}{u}\left\|\begin{array}[]{l}{(0,1)(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)$		(177)

valid for $\mathrm{Re}\, s>0$ , $C>0$ , $|\arg z|<\frac{1}{2}C\pi$ and $\min _{{1\leq j\leq m}}\mathrm{Re}\,(b_{j}/B_{j})>-1$ .

[page 125, §1]
[125.1.1] The definite integral found in [59, 2.25.2.2.]

	$\displaystyle\int _{0}^{y}x^{{\beta-1}}(y-x)^{{\gamma-1}}H^{{m,n}}_{{p,q}}\left(Cx^{\delta}(y-x)^{\eta}\left\|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)\mbox{\rm d}x$
	$\displaystyle=y^{{\beta+\gamma-1}}H^{{m,n+2}}_{{p+2,q+1}}\left(Cy^{{\delta+\eta}}\left\|\begin{array}[]{l}{(1-\beta,\delta),(1-\gamma,\eta),(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q}),(1-\beta-\gamma,\delta+\eta)}\end{array}\right.\right)$		(178)

contains as a special case the fractional Riemann-Liouville integral [58, (2.7.13)]

$\displaystyle\mbox{\rm I}^{{\alpha}}_{{0+}}H^{{m,n}}_{{p,q}}(y)$	$\displaystyle=\frac{1}{\Gamma(\alpha)}\int _{0}^{y}(y-x)^{{\alpha-1}}H^{{m,n}}_{{p,q}}\left(x\left\|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)\mbox{\rm d}x$
	$\displaystyle=y^{\alpha}H^{{m,n+1}}_{{p+1,q+1}}\left(y\left\|\begin{array}[]{l}{(0,1),(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q}),(-\alpha,1)}\end{array}\right.\right)$		(179)

valid if $\min _{{1\leq j\leq m}}\mathrm{Re}\,(b_{j}/B_{j})>0$ . [125.1.2] The fractional Riemann-Liouville derivative is obtained from this formula by analytic continuation to $\alpha<0$ .

4.4 Series Expansions

[125.2.1] The $H$ -functions may be represented as the series [56, 57, 58, 59]

$\displaystyle H^{{m,n}}_{{p,q}}\left(z\left\|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=\sum _{{i=1}}^{m}\sum _{{k=0}}^{\infty}c_{{ik}}\frac{(-1)^{k}}{k!B_{i}}z^{{(b_{i}+k)/B_{i}}}$		(180a)
where
$\displaystyle c_{{ik}}=\frac{\displaystyle\prod^{m}_{{\substack{j=1\\ j\neq i}}}\Gamma(b_{j}-(b_{i}+k)B_{j}/B_{i})\prod^{n}_{{j=1}}\Gamma(1-a_{j}+(b_{i}+k)A_{j}/B_{i})}{\displaystyle\prod^{q}_{{j=m+1}}\Gamma(1-b_{j}+(b_{i}+k)B_{j}/B_{i})\prod^{p}_{{j=n+1}}\Gamma(a_{j}-(b_{i}+k)A_{j}/B_{i})}$		(180b)

whenever $D\geq 0$ , ${\mathcal{L}}$ is as in (158) or (159) and the poles in $\mathbb{P}_{b}$ are simple. [125.2.2] Similarly

$\displaystyle H^{{m,n}}_{{p,q}}\left(z\left\|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=\sum _{{i=1}}^{n}\sum _{{k=0}}^{\infty}c_{{ik}}\frac{(-1)^{k}}{k!A_{i}}z^{{-(1-a_{i}+k)/A_{i}}}$		(181a)
[page 126, §0] where
$\displaystyle c_{{ik}}=\frac{\displaystyle\prod^{n}_{{\substack{j=1\\ j\neq i}}}\Gamma(1-a_{j}-(1-a_{i}+k)A_{j}/A_{i})\prod^{m}_{{j=1}}\Gamma(b_{j}+(1-a_{i}+k)B_{j}/A_{i})}{\displaystyle\prod^{p}_{{j=n+1}}\Gamma(a_{j}+(1-a_{i}+k)A_{j}/A_{i})\prod^{q}_{{j=m+1}}\Gamma(1-b_{j}-(1-a_{i}+k)B_{j}/A_{i})}$		(181b)

whenever $D\leq 0$ , ${\mathcal{L}}$ is as in (158) or (160) and the poles in $\mathbb{P}_{a}$ are simple.

Author:	R. Hilfer
originally published in:	Applications of Fractional Calculus in Physics, World Scientific, Singapore, page 87-130 (2000), ISBN 981-02-3457-0
originally published:	March, 2000

	$\displaystyle{\mathcal{L}}\left\{ H^{{m,n}}_{{p,q}}(z)\right\}(u)=\int _{0}^{\infty}e^{{-ux}}H^{{m,n}}_{{p,q}}\left(x\left\|\begin{array}[]{l}{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)\mbox{\rm d}x$
	$\displaystyle=H^{{n+1,m}}_{{q,p+1}}\left(u\left\|\begin{array}[]{l}{(1-b_{1}-B_{1},B_{1}),\ldots,(1-b_{q}-B_{q},B_{q})}\\ {(0,1)(1-a_{1}-A_{1},A_{1}),\ldots,(1-a_{p}-A_{p},A_{p})}\end{array}\right.\right)$
	$\displaystyle=\frac{1}{u}H^{{m,n+1}}_{{p+1,q}}\left(\frac{1}{u}\left\|\begin{array}[]{l}{(0,1)(a_{1},A_{1}),\ldots,(a_{p},A_{p})}\\ {(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)$		(177)