SynopsisExplicit orthogonality relations are found for the associated Laguerre and Hermite polynomials. One consequence is the construction of the [n − 1/n] Padé approximation to Ψ(a + 1, b; x)/Ψ(a, b; x), where Ψ(a, b; x) is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity.

Once on the forefront of mathematical research in America, the asymptotics of the solutions of linear recurrence equations is now almost forgotten, especially by the people who need it most, namely combinatorists and computer scientists. Here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her (or his) asymptotics tool kit.

In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c, for arbitrary real c ≧ 0. One expression allows us to give in closed form the [n — 1/n] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19]. (We actually give only the [n — 1/n] Padé element, although other cases are easily constructed, see [19] for details.)We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.

Clavaminate synthase is an Fe(2+)-, O2-, and alpha-ketoglutarate-dependent oxygenase that catalyzes three transformations in the biosynthesis of the important beta-lactamase inhibitor clavulanic acid. The genes from Streptomyces clavuligerus encoding two isoenzymes of clavaminate synthase have been over-expressed in Escherichia coli to give soluble proteins whose reactions, kinetic properties, and molecular masses are in excellent agreement with the wild-type isozymes. Preliminary investigation of the active site of clavaminate synthase was undertaken using diethyl pyrocarbonate and N-ethylmaleimide. Each was inhibitory to catalytic activity. Protection from inactivation in the presence of these reagents by Fe2+, O2, and alpha-ketoglutaric acid was thwarted by the rapid self-inactivation of the enzyme in the absence of substrate. However, protection was achieved when Co2+, a potent competitive inhibitor of clavaminate synthase 2 with respect to Fe2+, was substituted. This is consistent with the presence of histidine and cysteine, respectively, at or near the active site and possibly involved in iron binding. In the course of constructing the expression vector, a simply applied general error analysis of the polymerase chain reaction was formulated to calculate the proportion of correctly replicated DNA and guide the design of experiments using this method.

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