Indefinite Integration of the Gamma Integral
and Related Statistical Applications

7.1 Supporting Files

M1 (h.m)
Description: calculate the "$h$" function.
Input arguments: h($s$,$c$,$n_{2}$); argument $s$ is the base parameter; $c$ is the power parameter; $n_{2}$ is the number of summation terms for $\sum\limits_{s=0}^{\infty }{h_{s}^{-c}}$ when $s$ is a negative integer.

M2 (h1.m)
Description: calculate the "$h$" function with arbitrary precision.
Input arguments: h($s$,$c$,$pre$); argument $s$ is the base parameter; $c$ is the power parameter; $pre$ is the level of precision (number of singificant digits in matlab's vpa function).

M3 (difference.m)
Description: calculate an $n$th-order forward difference for the "$h$" function.
Input arguments: difference($s$,$c$,$order$); $s$ is the base parameter; $c$ is the power parameter; $order$ is the order of the forward difference.

M4 (hinv.m)
Description: calculate the inverse "$h$" function by numerical analysis.
Input arguments: hinv($k$,$s$); $k$ is the value of the "$h$" function; $s$ is base parameter.

M5 (f1expo.m)
Description: specify the exponential integral function of the first order.
Input arguments: f1expo($x$); $x$ is the integral variable.

M6 (f5expo.m)
Description: specify the exponential integral function of the fifth order.
Input arguments: f5expo($x$); $x$ is the integral variable.

M7 (f10expo.m)
Description: specify the exponential integral function of the tenth order.
Input arguments: f10expo($x$); $x$ is the integral variable.

M8 (hbetainc.m)
Description: evaluate the incomplete beta function by using the "$h$" functions.
Input arguments: hbetainc($x$,$a$,$b$,$nbeta$,$pre$); $x$ is the random variable; $a$ is parameter $\alpha$; $b$ is parameter $\beta$; $nbeta$ is the number of expansions for calcutaing the incomplete beta function; $pre$ is the level of precision (number of singificant digits in matlab's vpa function).

M9 (hmQ.m)
Description: estimate the Marcum Q-function in "$h$" forms.
Input arguments: hmQ($a$,$b$,$m$); $a$ is parameter $a$; $b$ is parameter $b$; $m$ is parameter $M$.

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