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A very general set of orthogonal polynomials with five free par ameter s is given explicitly, the orthogo nality relation is proved and the three term recurrence relatio n is found . AlAS (lAOS ) Subject Classification:33A65

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ω-Pentadecalactone (PDL) was copolymerized with lactones of varying sizes (6-, 7-, 9-, and 13-membered rings) in order to characterize the properties of PDL copolymers throughout the lactone range for copolymerizations catalyzed by magnesium 2,6-di-tertbutyl-4-methylphenoxide (Mg(BHT) 2 (THF) 2 ). Kinetics of the copolymerization reactions were studied using quantitative 13 C NMR spectroscopy, which revealed that the polymerization of the smaller, strained lactone monomer occurred rapidly before the incorporation of PDL into the polymer. Furthermore, all polymers were randomly sequenced as a consequence of transesterification side reactions that occurred throughout polymerization. The copolymers were all shown to cocrystallize to produce polymers with melting and crystallization temperatures that displayed a linear relationship with respect to monomer ratio. Differences in degradation behavior of the smaller lactones enabled the synthesis of PDL copolymer materials that displayed independently controllable thermal and degradation properties.

Much work has been directed to the design of complex single-site catalysts for ring-opening polymerization (ROP) to enhance both activity and selectivity. More simply, however, cooperative effects between Lewis acids and organocatalytic nucleophiles/Lewis bases provide a powerful alternative. In this study we demonstrate that the combination of N-heterocyclic carbenes, 1,8-diazabicycloundec-7-ene (DBU) and 4-dimethylaminopyridine (DMAP) with simple Lewis acids enables the ROP of the macrolactone pentadecalactone in a rapid and efficient manner. Remarkably, regardless of the nature of the nucleophile, the order of activity was observed to be MgX2 ≫ YCl3 ≫ AlCl3 and MgI2 > MgBr2 > MgCl2 in every case. The minimal influence of the organobase on polymerization activity allows for the use of simple and inexpensive precursors. Furthermore, extension of the study to other cyclic (di)ester monomers reveals the choice of Lewis acid to lead to monomer selective ROP activity and hence control over copolymer composition by choice of Lewis acid. This approach could lead to the realization of complex polymer structures with tunable physical properties from simple catalyst combinations.