Advice for research mathematicians | Bernoulli numbers and Faulhaber's sums of powers | WIld Egg

The Bernoulli numbers are an intriguing family of rational numbers that arise in many areas of analysis. We introduce them in the context of J. Faulhaber's formulas for sums of powers of natural numbers, which in fact give us another important family of polynomials or polynumbers.

These numbers will also figure importantly in the family of Bernoulli polynomials that we will have a look at in a subsequent video.

Video Contents:

00:02 - Introduction and Number Theory Journey
09:29 - Unpredictability of the Bernoulli Sequence
10:11 - Variants of the First Bernoulli Number
19:14 - Connection to Faulhaber Polynomials and Bernoulli’s Formula
22:05 - Notation and Curiosities in Bernoulli’s Work
26:05 – Extending Faulhaber Polynomials via Matrix Expansion
28:11 – Generating Larger Pascal-Harriet Matrices and Their Inverses
31:55 – Validating the Correct Formula by Summation Tests
33:50 – Conclusion and Recommendations

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