## Seminar in Lincoln: Professor Peter Larcombe (University of Derby)

**Professor Peter J. Larcombe (Dept. Computing and Mathematics, University of Derby) ****will speak in the Algebra seminar on Wednesday, 23 November 2016, at 2:00pm (room JBL0C05, JBL Building, which is building 22 on the uni map). He will talk about ***Catalan Numbers and Catalan Polynomials: An Overview of Some Work in Discrete Mathematics.*

__Abstract: __In this talk I will present some results from my work in discrete mathematics. Integer sequences whose (ordinary) generating functions satisfy a quadratic equation are common, and in any sequence instance the functional coefficients of the governing equation characterise a namesake family of polynomials. I will give some initial results for the general class of polynomials and then – in relation to the particular family of Catalan polynomials – present the algebraic execution of some numeric root finding schemes (including the classic Newton-Raphson and Halley algorithms) that are found to deliver non-linear polynomial identities; such identities – termed ‘auto-identities’, can be produced by computer for any polynomial family in the general polynomial class. On a different topic, it is found that a particular suite of trigonometric power series identities (whose origins date back to the some geometric models of the 17th century) provides fruitful ground for evaluations of sums of exponentiated multiples of Catalan number linear combinations – one or two different methods can be applied to achieve some pleasing results, and generalised.

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