UDC 517.5 + 517.6

MATHEMATICS

V. M. KALININ

ANALOGS OF STIRLING’S FORMULA

(Presented by Academician Yu. V. Linnik on 9 X 1967)

Stirling’s formula gives an asymptotic expansion of the gamma function \(\Gamma(1+x)\) as \(x \to \infty\) in inverse powers of \(x\):

\[ \Gamma(1+x)=\sqrt{2\pi x}\left(\frac{x}{e}\right)^x \exp\left\{\sum_{j=1}^{\nu-1}\frac{B_{j+1}}{j(j+1)x^j} +O\left(\frac{1}{x^\nu}\right)\right\}, \]

\[ \Gamma(1+x)=\sqrt{2\pi x}\left(\frac{x}{e}\right)^x \left\{1+\sum_{j=1}^{\nu-1}\frac{b_j}{x^j} +O\left(\frac{1}{x^\nu}\right)\right\}, \]

\[ \frac{1}{\Gamma(1+x)}=\frac{1}{\sqrt{2\pi x}}\left(\frac{e}{x}\right)^x \left\{1+\sum_{j=1}^{\nu-1}(-)^j\frac{b_j}{x^j} +O\left(\frac{1}{x^\nu}\right)\right\}, \]

where the coefficients \(b_j\) can be found from the recurrence relations

\[ b_j=\frac{1}{j}\sum_{k=0}^{j-1}\frac{b_k B_{j-k+1}}{j-k+1}, \qquad b_0=1. \]

Here \(B_{j-k+1}\) are Bernoulli numbers. (Somewhat different recurrence equations are indicated in (1).) The Bernoulli numbers are easily computed recursively:

\[ B_j=(-)^j j\sum_{k=0}^{j-1}\frac{B_k C_{j-1}^k}{j-k+1}, \qquad B_1=-\frac{1}{2}. \]

In applications (for example, in probability theory) it is often convenient to expand the gamma function in powers of a quantity different from \(x\). The following theorems introduce into Stirling’s formula an arbitrary parameter \(\theta\), by the choice of which one can obtain the necessary expansions.

Theorem 1. For \(x>-1\), \(\nu=2,3,\ldots\), and arbitrary \(\theta>-x\)

\[ \Gamma(1+x)=\sqrt{2\pi}\,(x+\theta)^{x+1/2} \exp\left\{-(x+\theta)+ \sum_{j=1}^{\nu-1}\frac{B_{j+1}(\theta)}{j(j+1)(x+\theta)^j} +R_\nu\right\}, \]

where

\[ R_\nu=-\frac{1}{\nu}\int_0^1 B_\nu(t)\zeta(\nu,x+t)\,dt +\frac{1}{\nu}\int_1^\theta \frac{B_\nu(t)}{(x+t)^\nu}\,dt, \]

\[ \zeta(\nu,x)=\sum_{k=1}^{\infty}\frac{1}{(x+k)^\nu}, \]

\(B_{j+1}(\theta)\) are Bernoulli polynomials. They may be found, for example, from the following recurrence formulas:

\[ (-)^j\frac{B_j(\theta)}{j!} = \sum_{k=0}^{j-1}(-)^k\frac{B_k(\theta)}{k!} \frac{(\theta-1)^{j-k+1}-\theta^{j-k+1}}{(j-k+1)!}, \qquad B_0(\theta)=1. \]

The proof of Theorem 1 is close to the proof of Lemma 1 of paper (2); the difference is that in Taylor’s formula, on which the summation theorem is based, the remainder term is taken in integral form.

Theorem 2. As \(x\to\infty\), \(\nu=1,2,\ldots\), uniformly with respect to arbitrary \(\theta=O(1)\), the expansions

\[ \Gamma(1+x)=\sqrt{2\pi}\,(x+\theta)^{x+1/2} \exp\left\{-(x+\theta)+\sum_{j=1}^{\nu-1} \frac{B_{j+1}(\theta)}{j(j+1)(x+\theta)^j} +O\left(\frac{1}{(x+\theta)^\nu}\right)\right\}, \]

\[ \Gamma(1+x)=\sqrt{2\pi}\,(x+\theta)^{x+1/2}e^{-(x+\theta)} \left\{1+\sum_{j=1}^{\nu-1}\frac{b_j(\theta)}{(x+\theta)^j} +O\left(\frac{1}{(x+\theta)^\nu}\right)\right\}, \]

\[ \frac{1}{\Gamma(1+x)}= \frac{e^{x+\theta}}{\sqrt{2\pi}\,(x+\theta)^{x+1/2}} \left\{1+\sum_{j=1}^{\nu-1}(-1)^j \frac{b_j(1-\theta)}{(x+\theta)^j} +O\left(\frac{1}{(x+\theta)^\nu}\right)\right\}, \]

hold, where \(b_j(\theta)\) are polynomials of degree \(2j\), satisfying the recurrence relations

\[ b_j(\theta)=\frac{1}{j}\sum_{k=0}^{j-1} \frac{b_k(\theta)B_{j-k+1}(\theta)}{j-k+1}, \qquad b_0(\theta)=1, \]

\[ \Delta b_j(\theta)=\theta b_{j-1}(1+\theta). \]

In particular, from Theorem 2 we find, as \(x\to\infty\),

\[ \Gamma(1+x)=[x]!\,x^{\{x\}} \exp\left\{\sum_{j=1}^{\nu-1} \frac{B_{j+1}-B_{j+1}(\{x\})}{j(j+1)x^j} +O\left(\frac{1}{x^\nu}\right)\right\}, \]

(here \([x]\) and \(\{x\}\) are the integer and fractional parts of \(x\)).

As \(n\to\infty\) and \(x=O(1)\), the following equality holds uniformly with respect to \(x\):

\[ \Gamma(x)= \frac{\sqrt{2\pi}\,n^{\,n+x+1/2}e^{-n}} {x(x+1)\cdots(x+n)} \exp\left\{\sum_{j=1}^{\nu-1} \frac{B_{j+1}(-x)}{j(j+1)n^j} +O\left(\frac{1}{n^\nu}\right)\right\}. \]

Theorem 3. As \(x\to\infty\), uniformly with respect to arbitrary \(\theta=O(\sqrt{x})\) and arbitrary \(a=O(1)\), the expansions

\[ \Gamma(1+x)=\sqrt{2\pi}\,(x+\theta)^{x+1/2} \exp\left\{-(x+\theta)+\frac{y^2}{2} +\sum_{j=1}^{\nu-1} \frac{v_j(y,a)}{(\sqrt{x+\theta})^j} +O\left(\frac{1}{(\sqrt{x+\theta})^\nu}\right)\right\}, \]

\[ \Gamma(1+x)=\sqrt{2\pi}\,(x+\theta)^{x+1/2}e^{-(x+\theta)-y^2/2} \left\{1+\sum_{j=1}^{\nu-1} \frac{w_j(y,a)}{(\sqrt{x+\theta})^j} +O\left(\frac{1}{(\sqrt{x+\theta})^\nu}\right)\right\}, \]

\[ \frac{1}{\Gamma(1+x)}= \frac{e^{x+\theta-y^2/2}}{\sqrt{2\pi}\,(x+\theta)^{x+1/2}} \left\{1+\sum_{j=1}^{\nu-1}(-i)^j \frac{w_j(iy,1-a)}{(\sqrt{x+\theta})^j} +O\left(\frac{1}{(\sqrt{x+\theta})^\nu}\right)\right\}, \]

hold, where

\[ y=\frac{\theta-a}{\sqrt{x+\theta}}, \]

\[ v_j(y,a)= \sum_{k=0}^{1+[j/2]} \frac{B_k(a)C_{j-k+2}^{\,k}y^{j-2k+2}} {(j-k+1)(j-k+2)}, \]

and \(w_j(y,a)\) are polynomials of degree \(3j\) in \(y\) of parity \(j\), satisfying the recurrence relations

\[ w_j(y,a)=\sum_{k=0}^{j-1} \left(1-\frac{k}{j}\right)w_k(y,a)v_{j-k}(y,a), \qquad w_0(y,a)=1. \]

(The imaginary unit \(i\) enters the expression \((-i)^j w_j(iy,1-a)\) only in even powers.)

Theorems 2 and 3 are direct consequences of Theorem 1. All the assertions formulated can also be given in a complex version. Thus, Theorem 2 remains valid if \(x\) is replaced by a complex variable \(z\) as \(|z|\to\infty\), uniformly with respect to an arbitrary complex quantity \(\theta=O(1)\) and with respect to \(|\arg z|\le \pi-\varepsilon\). Theorem 3 holds when \(x\) is replaced by \(z\), uniformly with respect to arbitrary complex quantities \(\theta=O(\sqrt[\nu]{z})\), \(a=O(1)\), and with respect to \(|\arg z|\le \pi-\varepsilon\). This makes it possible, in particular, to describe the asymptotic behavior of \(\Gamma(1+z)\) for \(z=x+iy\). For example, one may state the following result:

Theorem 4. For \(z=x\pm iy\), \(y\to\infty\), the following equalities hold uniformly with respect to \(x=O(1)\):

\[ \Gamma(1+z)=\sqrt{2\pi}\,y^{x+1/2}\exp\left\{-\frac{\pi}{2}y\pm i\left[\frac{\pi}{2}\left(x+\frac12\right)+y(\ln y-1)\right]+ \sum_{j=1}^{\nu-1}\frac{(\mp i)^j B_{j+1}(-x)}{j(j+1)y^j} +O\left(\frac1{y^\nu}\right)\right\}, \]

\[ \Gamma(1+z)=\sqrt{2\pi}\,y^{x+1/2}\exp\left\{-\frac{\pi}{2}y\pm i\left[\frac{\pi}{2}\left(x+\frac12\right)+y(\ln y-1)\right]\right\}\times \]

\[ \times\left\{1+\sum_{j=1}^{\nu-1}(\mp i)^j\frac{b_j(-x)}{y^j} +O\left(\frac1{y^\nu}\right)\right\}. \]

From the complex version of Theorem 3 one can determine the asymptotic behavior of \(\Gamma(1+x\pm iy)\) as \(y\to\infty\) and \(x=O(\sqrt[\nu]{y})\). For example, we immediately obtain

\[ \left|\Gamma(1+x\pm iy)\right| = \sqrt{2\pi}\,y^{x+1/2}e^{-\frac{\pi}{2}y} \left\{1+O\left(\frac1{\sqrt[\nu]{y}}\right)\right\} \]

uniformly with respect to \(x=O(\sqrt[\nu]{y})\).

We note that, instead of the expansions of \(\Gamma(1+x)\) and \(\Gamma(1+z)\), one may everywhere write, with some obvious changes, the expansions of \(\Gamma(x)\) and \(\Gamma(z)\).

In the last two theorems the behavior of a frequently occurring ratio is described.

Theorem 5. As \(|z|\to\infty\), uniformly with respect to an arbitrary complex \(\theta=O(1)\) and with respect to \(|\arg z|\le \pi-\varepsilon\), the following expansions hold:

\[ \frac{\Gamma(z+\theta)}{\Gamma(z)} = z^\theta \exp\left\{ \sum_{j=1}^{\nu-1}(-)^j \frac{B_{j+1}-B_{j+1}(\theta)}{j(j+1)z^j} +O\left(\frac1{z^\nu}\right) \right\}, \]

\[ \frac{\Gamma(z+\theta)}{\Gamma(z)} = z^\theta \left\{ 1+\sum_{j=1}^{\nu-1}\frac{C_\theta^{(j)}}{z^j} +O\left(\frac1{z^\nu}\right) \right\}, \]

\[ \frac{\Gamma(z)}{\Gamma(z+\theta)} = z^{-\theta} \left\{ 1+\sum_{j=1}^{\nu-1}(-)^j\frac{C_{1-\theta}^{(j)}}{z^j} +O\left(\frac1{z^\nu}\right) \right\}, \]

\[ C_\theta^{(j)} = \theta(\theta-1)\cdots(\theta-j) \sum_{k=0}^{j-1}g_{jk}(\theta-j-1)\cdots(\theta-j-k), \]

\[ g_{j0}=\frac1{j+1},\qquad g_{j,j-1}=\frac1{2^j j!},\qquad g_{jk}=\frac{(j+k)g_{j-1,k}+g_{j-1,k-1}}{j+k+1}, \quad k=1,\ldots,j-2. \]

(The values of the polynomials \(C_\theta^{(j)}\) for \(\theta=j+1,j+2,\ldots\) are the Stirling numbers of the first kind \(C_\theta^{(j)}\), and for \(\theta=-1,-2,\ldots\) the Stirling numbers of the second kind \(\mathfrak{S}_{-\theta}^{(j)}\).)

Theorem 6. As $|z|\to\infty$, uniformly with respect to an arbitrary complex $\theta=O(1)$ and with respect to $|\arg z|\leqslant \pi-\varepsilon$, the following expansions hold:
\[ \frac{\Gamma(z+\theta\sqrt z)}{\Gamma(z)} = z^{\theta\sqrt z}\exp\left\{ \frac{\theta^2}{2} + \sum_{j=1}^{\nu-1}(-)^j\frac{\alpha_j}{(\sqrt z)^j} + O\left(\frac{1}{(\sqrt z)^\nu}\right) \right\}, \]
\[ \frac{\Gamma(z+\theta\sqrt z)}{\Gamma(z)} = z^{\theta\sqrt z}e^{\theta^2/2} \left\{ 1+ \sum_{j=1}^{\nu-1}(-)^j\frac{\beta_j}{(\sqrt z)^j} + O\left(\frac{1}{(\sqrt z)^\nu}\right) \right\}, \]
\[ \frac{\Gamma(z)}{\Gamma(z+\theta\sqrt z)} = z^{-\theta\sqrt z}e^{-\theta^2/2} \left\{ 1+ \sum_{j=1}^{\nu-1}(-)^j\frac{\gamma_j}{(\sqrt z)^j} + O\left(\frac{1}{(\sqrt z)^\nu}\right) \right\}, \]
where
\[ \alpha_j = \sum_{k=0}^{[(j+1)/2]} (-)^k \frac{C_j^k B_k \theta^{\,j-2k+2}} {(j-k+1)(j-k+2)}, \]
\[ \beta_j = \sum_{k=0}^{j-1} \left(1-\frac{k}{j}\right)\beta_k\alpha_{j-k}, \qquad \beta_0=1, \]
\[ \gamma_j = -\sum_{k=0}^{j-1} \left(1-\frac{k}{j}\right)\gamma_k\alpha_{j-k}, \qquad \gamma_0=1. \]

Another analogue of Stirling’s formula is contained in work (2).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
2 X 1967

REFERENCES

1. E. Copson, Asymptotic Expansions, Moscow, 1966.
2. V. M. Kalinin, Theory of Probability and Its Applications, 12, no. 1, 24 (1967).