Orthogonal Polynomials and Special Functions Lecture 4: Christoffel-Darboux Kernel and Roots

This is the fourth video in a series of lectures intended to introduce talented undergraduate students to topics in orthogonal polynomials and special functions, with the aim of preparing for independent undergraduate mathematical research.
In this video we define the Christoffel-Darboux kernel and use it to prove the interlacing property of roots of orthogonal polynomials.

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Sorry but I didn't really get the step at 7:49, where did ck+1 came from, when i do the substraction my only matching terms are BnPk(x)Pk(y) which cancel out