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Classical orthogonal polynomials have widespread applications including in numerical integration, solving differential equations, and interpolation. Changing basis between classical orthogonal polynomials can affect the convergence, accuracy, and stability of solutions. We provide a general method for changing basis between any pair of classical orthogonal polynomials by using algebraic expressions called coefficient functions that evaluate to connection coefficients. The method builds directly on previous work on the change of basis groupoid. The scope has fifteen kinds of classical orthogonal polynomials including the classes of Jacobi, Gegenbauer and generalized Laguerre polynomials. The method involves the mappings to and from the monomials for these polynomial bases. Sixteen coefficient functions appear to be new for polynomials that do not have definite parity. We derive the remainder from seven sources in the literature. We give a complete summary of thirty coefficient functions. This enables change of basis to be defined algebraically and uniformly between any pair of classical orthogonal polynomial bases by using a vector dot product to compose two coefficient functions to give a third. A definition of Jacobi polynomials uses a basis of shifted monomials. We find a key new mapping between the monomials and Jacobi polynomials by using a general mapping between shifted monomials and monomials. It yields new mappings for the Chebyshev polynomials of the third and fourth kinds and their shifted versions., Comment: 25 pages

We show that the change of basis matrices of a set of $m$ bases of a finite vector space is a connected groupoid of order $m^2$. We define a general method to express the elements of change of basis matrices as algebraic expressions using optimizations of evaluations of vector dot products. Examples are given with orthogonal polynomials., Comment: 23 pages

We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$ where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring variants of Chebyshev polynomials of the first and second kinds, $T_n(x) \pm 1$ and $U_n(x) \pm 1$. We also show that, in general, there are no factorizations of variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x) \pm 1$ and $Y_n(x) \pm 1$ using minimal polynomials of $\cos(\frac{2��}{d})$., 9 pages

Spricht man von den gangigen Veranstaltungstypen auf deutschen Messe- und Kongressplatzen, so wird haufig zwischen drei Veranstaltungsgattungen unterschieden: Messen, Kongresse und sonstige Veranstaltungen/Events. Letztere bilden eine Art Sammelbegriff fur jene Veranstaltungen, die sich nicht eindeutig in den beiden ersten Kategorien abbilden lassen. Als Beispiele konnen v. a. Firmenveranstaltungen genannt werden, die entweder ihren Schwerpunkt z. B. in der Prasentation von Produkten oder Dienstleistungen haben oder schlichtweg Mischformen zwischen Messen und Kongressen sind. Hierzu zahlen viele Veranstaltungen aus der IT- und Kommunikationsbranche. Diese Dreiteilung des Veranstaltungsgeschafts greift, insbesondere bei den in diesem Beitrag im Zentrum stehenden Kongresszentren, zu kurz. Vielmehr ist es sinnvoller, die Unterscheidung grundsatzlich danach vorzunehmen, wer die Rolle des Veranstalters innehat. Konsequenterweise kommt dabei die Unterscheidung in Eigen- und Gastveranstaltungen zum Tragen.