REPRESENTATION OF THE GAUSSIAN FUNCTION IN A SPECIAL CASE BY ELEMENTARY FUNCTIONS

V. E. KORNILOV

(Submitted by the Department of Higher Mathematics)

In the article, cases are studied in which the Gaussian function

\[ F(a,b;c;z)=\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{k!(c)_k}z^k,\quad |z|<1, \]

\[ c\ne 0,-1,\ldots;\quad (a)_k=a(a+1)\ldots(a+k-1) \tag{1} \]

is represented by elementary functions.

Theorem. The hypergeometric series (1) represents elementary functions in the following cases:

1. For complex parameters \(a,b,c\) such that

1) \(a,b,c-a\) or \(c-b=0,-1,\ldots;\)

2) \(a,c=1,2,\ldots\)

the function (1) is elementary.

1. If the complex parameters \(a,b,c\) are such that two of the three numbers

\[ \frac{1}{2}-c,\quad a-b-\frac{1}{2},\quad a+b-c-\frac{1}{2} \]

are equal to \(0,\pm1,\ldots\), then the series (1) represents elementary functions.

1. If the parameter \(a=1,2,\ldots\) and the remaining \(b,c\) take rational values such that \(c-b\) or \(b=1,2,\ldots\), then in this case as well the function (1) will be elementary.

Proof. 1. 1) On the basis of the equalities

\[ F(-n,b;c;z)=\sum_{k=0}^{n}\frac{(-n)_k(b)_k}{k!(c)_k}z^k, \tag{2} \]

\[ F(a,b;c;z)=(1-z)^{-a}F\left(a,c-b,c;\frac{z}{z-1}\right) \tag{3} \]

the theorem is valid if \(a,b,c-a\) or \(c-b=0,-1,\ldots\)

2) For \(a=m,\ c=1,\ldots,m\), on the basis of equality (3) we obtain elementary functions. If \(a=m;\ c=m+1,m+2,\ldots\), then by virtue of the following equalities

\[ \left\{ \begin{aligned} F(m,b;c;z)={}&\sum_{\kappa=1}^{m-1}(-1)^\kappa C_{m-1}^{\kappa-1} \frac{(b+1-\kappa)_{m-1}}{(1)_{m-1}}\times\\ &\times\left[ \sum_{j=0}^{m-\kappa-1} \frac{(c-b-m+\kappa)_j\sqrt m}{(1-b-m-\kappa)_{j+1}} \frac{c-1}{(1-z)^{j+1}} \right] +\frac{(1-b)_{m-1}}{(1)_{m-1}}\times\\ &\times F\left(1-m,b-c+1;1+b-m;\frac{1}{1-z}\right)\cdot F(1,b;c;z), \end{aligned} \right. \tag{4} \]

\[ \left\{ \begin{aligned} F(1,b;c;z)={}&(-1)^{c-1}\frac{(c-1)!}{(1-b)_{c-1}} \left(\frac{1-z}{z}\right)^{c-1}\times\\ &\times\left[ \frac{1}{(1-z)^b} -\frac{1}{1-z}\sum_{\kappa=0}^{c-2} (-1)^\kappa\frac{(1-b)_\kappa}{\kappa!} \left(\frac{z}{1-z}\right)^\kappa \right] \end{aligned} \right. \tag{5} \]

the Gaussian function is represented by elementary functions.

1. Let, in particular, \(\frac12-c=a-b-\frac12=0\); then \(c=\frac12\),

\[ a=b+\frac12 \]

and series (1), on the basis of ([1], p. 110 (5)), is the following:

\[ F\left(b,b+\frac12;\frac12;z\right) =\frac12(1+\sqrt z)^{-2b}+\frac12(1-\sqrt z)^{-2b}. \tag{6} \]

If \(a-b-\frac12=a+b-c-\frac12=0\), then \(a=b+\frac12,\ c=2b\)

and series (1) is the following ([1], p. 110 (6)):

\[ F\left(b,b+\frac12;2b;z\right) =\frac{1}{\sqrt{1-z}} \left(\frac12+\frac12\sqrt{1-z}\right)^{1-2b}. \tag{7} \]

In the general case, if

\[ \frac12-c=\pm k,\quad a-b-\frac12=\pm m, \]

the values of the parameters \(c,a\), depending on the case, can be raised or lowered by an integer according to the recurrence formulas ([1], p. 110 (20)—(27)). Thus, function (1) is transformed into the sum of a polynomial and the product of a polynomial by function (6). Similarly, the formulas mentioned above are also applied to the case

\[ a-b+\frac12=\pm k,\quad a+b-c-\frac12=\pm m\ (m,k=1,2,\ldots). \]

Function (1) is transformed into the sum of a polynomial and the product of a polynomial by function (7). In the latter case, when

\[ \frac12-c=\pm k, \]

\[ a+b-c-\frac12=\pm (m,k=1,2,\ldots). \]

Here formula (3) is also applied, and function (1) is reduced to the sum of a polynomial and the product of a polynomial, a power function, and function (6).

1. A more general case of equality (4) is the equality [2]

\[ \begin{aligned} F(1+m,b;c;z) &= \sum_{k=1}^{m}(-1)^k C_m^{k-1}\frac{(b+1-k)_m}{m!}\times \\ &\quad \times \left[ \sum_{j=0}^{m-k} \frac{(c-b-m-1+k)_j\,c-1}{(-b-m+k)_{j+1}(1-z)^{j+1}} \right] +\frac{(1-b)_m(c-1)}{(1-z)^{b+1-c}}\times \\ &\quad \times \frac{z^{1-c}}{m!}\, F\!\left(-m,b-c+1;b-m;\frac{1}{1-z}\right) \int_{0}^{z}\frac{t^{c-2}\,dt}{(1-t)^{c-b}}, \quad c>1. \end{aligned} \tag{8} \]

If \(c-b\) or \(b=1,2,\ldots\), then for rational \(c\) and \(b\) the integral on the right-hand side of equality (8) is represented by elementary functions [2]. The theorem is proved.

The proof of equality (8) is analogous to that set forth in the article ([3], p. 17 ((14))).

REFERENCES

1. H. Bateman and A. Erdélyi. Higher Transcendental Functions. Moscow, Fizmatgiz, 1965.

2. V. E. Kornilov. Separation of the algebraic part of integrals of binomial differentials. Izv. TPI, Vol. 131, 1965.

3. V. E. Kornilov. Transformation into continued fractions of certain power series. Izv. TPI, Vol. 154, 1967.