Some Basic Properties of the Incomplete Gamma Function Ratio

Abstract

We define the incomplete gamma function ratio for positive $N$ by \begin{align*}\tag{1}P(N, b, X) &= \int^X_0 t^{N-1}e^{-bt} dt/ \int^\infty_0 t^{N-1}e^{-bt} dt \\ &\equiv \int^X_0 D(N, b, t) dt, \text{say},\\ \end{align*} where $N$ and $b$ are positive real numbers and $0 < X \leqq \infty$. In what follows, we generally use the notation $P_N$ to denote the same function, unless $b$ or $X$ are given some specific values. It should be noted that the notation $P(N, b, X)$ does not mean that the function is one of three variables, since we have $P(N, b, X) = P(N, 1, bX)$. The positive real number $b$ is only a scale factor. The function $P_N$ is of appreciable importance in probability and statistics and it is known as the gamma distribution and for $b = \frac{1}{2}$ as the chi-square distribution. It is also of importance in many other branches of applied mathematics. Like the incomplete beta distribution, it is also a special case of the more general confluent hypergeometric function. During the last decade, interest has been revived in the development, use and application of these functions and a number of publications, such as Erdelyi et al. (1953), Tricomi (1954) and Slater (1960), appeared. In these publications, the function treated is defined as the numerator in the middle part of (1) with $b = 1$, and the definition is generally extended for all complex values of $N$ such that $R(N)$ is not a non-positive integer. Many of the properties of $P_N$ are derived in these publications. Bancroft (1949) derived some new properties of the incomplete beta function (particularly, recurrence relations) from the more general properties of the parent hypergeometric function. Similar methods may also be used to derive new properties of $P_N$ of use in statistical work. In this paper, however, we derive such properties directly from the well-known difference-differential properties of $P_N$ given by \begin{align*}\tag{2}(\partial^r/\partial X^r) P_N &= (-b)^r\Delta^rP_{N-r}, \\ &= (\partial^{r-1}/\partial X^{r-1})D(N, b, X),\quad N > r,\\ \end{align*} where $r = 0, 1, 2, 3, \cdots$ and where \begin{equation*}\tag{3}\Delta^{r+1} P_T = \Delta^r(\Delta P_T) = \Delta^r(P_{T+1} - P_T)\end{equation*} are the advancing finite differences of $P_N$ with respect to $N$ and with a unit difference interval. Property (2) for $r = 1$ may be used, in fact, to define the function $P_N$ itself, as in Milne-Thomson (1933). This property was also used in 1947 by Khamis in an unpublished Ph.D thesis (1950) to derive, inter alia, a series expansion in terms of chi-square integrals to approximate to statistical distribution functions and also to formulate a computational method for the tabulation of the chi-square distribution itself. This method was later used by Hartley and Pearson (1950) in computing a five decimal table for this distribution. The chi-square series expansion was later generalized by Khamis (1960a) into an expansion in terms of incomplete gamma function ratios and the computational method was employed by Khamis, (1964a) and (1965) to prepare a six decimal table of the chi-square integral and a master ten decimal table, for very fine $N$ and $X$ intervals, of the incomplete gamma function (1) for $b = \frac{1}{2}$ (a description of this table is given by Khamis (1964b). In this paper we make further use of the property (2) to derive other new and useful properties of the function (1). In Section 2 we derive a simple recurrence relation for $P(T, b, X)$ for $T = N, N + 1$ and $N + 2$. In Section 3 we derive new $N$-wise sum formulae as consequences of this recurrence relation. We then give in Section 4 an $X$-wise sum formula based on the Taylor expansion of (2) in the neighbourhood of $X$. This is made possible by extending the definition of $P(N, b, X)$ for $N \leqq 0$, and thus enabling the removal of the condition $N > r$ in (2) above. Other examples of the use of property (2) are given in Section 5 where in particular, the Laguerre polynomials are expressed in terms of the differences of the function $P(N, b, X)$. Section 5 contains also examples of some numerical applications of the previous results. The importance of the properties derived here and the simplicity inherent in such derivations due to the nature of property (2) are further enhanced by work recently carried out by Wise (1950) and developed further by H. O. Hartley and E. J. Hughes (in process of publication) where the incomplete gamma function ratio is shown to provide quite satisfactory approximations to the incomplete beta function ratio and to many other statistical functions. We note finally, that the notation used in (1) is only one of many notations used by different authors. The references given above include most of the different notations used for the function $P_N$ and other related functions.