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Taking reciprocals in the finite product defining the gamma function givesLet $H_n={\sum }_{k=1}^n{k}^{-1}$. Sinceand $H_n-\log n\to \gamma$, where $\gamma$ is the Euler--Mascheroni constant, passage to the limit gives the Weierstrass product for the reciprocal gamma function

The logarithmic derivative of this identity isSince $k^{-1}-(z+k{)}^{-1}=z/[k(z+k)]$, the digamma function therefore satisfies

For real $z>0$, termwise differentiation gives the trigamma functionThus $\psi$ is strictly increasing on the positive real axis. At the two positive integers needed here, telescoping givesThe intermediate value theorem supplies a zero in $(1,2)$, and strict increase makes it unique. This is the positive zero of the digamma function.

With population $n$, the transition diagram has the two outgoing arrowsThus this is a batch-birth linear-death process. If $P_n(t)=\mathbb{P}(N_t=n)$ and $P_j=0$ for $j<0$, probability enters state $n$ from $n-3$ by a birth and from $n+1$ by a death. Hence the master equation is

The Markov jump-process generator isFor $f(n)=n$, this gives $Ln=3\lambda -\beta n$. Therefore the expected value $m(t)=\mathbb{E}[N_t]$ satisfiesso

For $f(n)=n^2$,Writing $s=\mathbb{E}[N_t^2]$ and using the generator identity givesThe variance $v=s-m^2$ consequently obeysIf $v(0)=v_0<\infty$, substitution of the formula for $m$ yieldsAll transient terms vanish, and the moments of a batch-birth linear-death process therefore give

For $X=x$, the last flip must be the $r$th head. Among the preceding $x+r-1$ flips, exactly $x$ are tails and $r-1$ are heads. There are $\left(\genfrac{}{}{0ex}{}{x+r-1}{x}\right)$ choices for their positions, and every resulting sequence has probability $(1-p{)}^x{p}^r$. The negative binomial stopping argument therefore givesso $X$ has the failures-before-the-$r$th-success negative binomial distribution.

To put the mass function into exponential family form, setSince $p=1-e^{\theta }$,whereThus the natural parameter of an exponential family is $\theta$, and the Fisher-Neyman factorization theorem shows that $T(X)=X$ is a sufficient statistic. This is the negative binomial exponential family.

The exponential-family derivative identities now giveand