Taking reciprocals in the finite product defining the gamma function gives
Let . Since
and , where is the Euler--Mascheroni constant, passage to the limit gives the Weierstrass product for the reciprocal gamma function
The logarithmic derivative of this identity is
Since , the digamma function therefore satisfies
For real , termwise differentiation gives the trigamma function
Thus is strictly increasing on the positive real axis. At the two positive integers needed here, telescoping gives
The intermediate value theorem supplies a zero in , and strict increase makes it unique. This is the positive zero of the digamma function.
With population , the transition diagram has the two outgoing arrows
Thus this is a batch-birth linear-death process. If and for , probability enters state from by a birth and from by a death. Hence the master equation is
The Markov jump-process generator is
For , this gives . Therefore the expected value satisfies
so
For ,
Writing and using the generator identity gives
The variance consequently obeys
If , substitution of the formula for yields
All transient terms vanish, and the moments of a batch-birth linear-death process therefore give
For , the last flip must be the th head. Among the preceding flips, exactly are tails and are heads. There are choices for their positions, and every resulting sequence has probability . The negative binomial stopping argument therefore gives
so has the failures-before-the-th-success negative binomial distribution.
To put the mass function into exponential family form, set
Since ,
where
Thus the natural parameter of an exponential family is , and the Fisher-Neyman factorization theorem shows that is a sufficient statistic. This is the negative binomial exponential family.
The exponential-family derivative identities now give
and

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact