[Revisited] Using a Computer to Derive Every* Possible Identity

Please check out the source of this video: https://www2.math.upenn.edu/~wilf/AeqB.html Original video: https://youtu.be/0LFg5dvPOoc An informal overview of how to use a computer to solve the problem of finding closed forms of hypergeometric identities, covering 2 of the 4 major algorithms: Sister Celine's method and Gosper's algorithm. The video goes back over the main ideas in the first video, covering hypergeometric series, telescoping series, Sister Celine's method, Gosper's algorithm (as well as Gosper canonical form and the method of undetermined coefficients), and the Wilf-Zeilberger proof algorithm and certificates. This time, there is a more structured high-level view to better understand how all of the algorithms fit together in context. Petition for a math video you would like to see: https://www.ablebees.com/team/zhuli My Patreon: https://www.patreon.com/zhulimath/ 00:00 Prerequisites and Overview 01:01 Chapter 1: Overview of Hypergeometric Functions 01:05 1-1: What does hypergeometric mean? 01:48 1-2: High-level summary 02:07 1-3: Recurrence definition 02:46 1-4: General hypergeometric term formula 06:20 1-5: F(n+i, k+j) / F(n,k) is always rational 07:15 1-6: Telescoping and reduction strategy 08:44 1-7: Summary of chapter 1 09:18 Chapter 2: Sister Celine's Method 09:21 2-1: Overview (and note about "coefficients") 11:22 2-2: Example (and note about "compact support") 14:42 2-3: Fundamental Theorem of Hypergeometric Functions 15:48 2-3a: Highest degree / number of equations 18:37 2-3b: Number of terms / number of unknowns 19:16 2-4: A faster way? 21:02 Chapter 3: Gosper's Algorithm 21:06 3-1: Overview 22:28 3-2: Hypergeometric to rational 23:39 3-3: Rational to polynomial 24:03 3-4: Gosper canonical form 25:37 3-4a: Additional constraints / alternate intuition 28:44 3-5: Proof that recurrence is polynomial 31:11 3-6: Method of undetermined coefficients 33:00 3-6a: Degree analysis 36:43 3-7: Example 39:14 Chapter 4: The WZ Method 39:17 4-1: Review 40:53 4-2: Proof certificate overview 41:41 4-3: The WZ method 42:46 4-4: Example (applying the proof certificate) 43:49 4-5: Obtain the proof certificate (Gosper's algorithm) 44:24 4-6: Example (finding the proof certificate) 46:12 Outro 46:42 Petition for me to make a video of your choice! Special thanks to Anakin Dey (https://anakin.phd/) for revision and proofreading. Music Credit: Hara Noda / Aligned Ever So Blue / Redolent Franz Gordon / Sea Arc Eight Helmut Schenker / Super 8 Sayuri Hayashi Egnell / Nagi Sayuri Hayashi Egnell / Flowers in the Mirror Lo Mimieux / Nod William Claeson / Hoppfull Christophe Gorman / Octagon William Benckert / Meant to Love Franz Gordon / Tea at Marlows Johannes Bornlof / Be My Always Ever So Blue / Threads Gavin Luke / Homecoming courtesy of www.epidemicsound.com Related tags: WZ pair, WZ method, computer algebra systems, Gosper's algorithm, Sister Celine's method, proof certificate, hypergeometric functions