Recursion formula, proof

Polya 3, 4, 5. The last paper contains a direct proof for the equivalence of the recursion formulas based on combinatorial considerations and on the functional equation (4). 

Rather similar was the paper [PolG36a] which also derives asymptotic formulae for the number of several kinds of chemical compounds, for example the alcohols and benzene and naphthalene derivatives. Unlike the paper previously mentioned, this one gives proofs of the recursion formulae from which the asymptotic results are derived. A third paper on this topic [PolG36] covers the same sort of ground but ranges more broadly over the chemical compounds. Derivatives of anthracene, pyrene, phenanthrene, and thiophene are considered as well as primary, secondary, and tertiary alcohols, esters, and ketones. In this paper Polya addresses the question of enumerating stereoisomers -- a topic to which we shall return later. 

Bessel functions have many interesting properties that will be presented here without proof, e.g. the recursion formula... 

The recursion formula, the normalizing constant, and the proof for the orthogonality relation among eigenvectors are given in Appendix IV. And pi r) is given such that... 

The component of the eigenvector can be obtained by the same argument as in the previous cases i. e., the wth component of the eigenvector is proportional to, S , and is uniquely determined by the orthogonalization constant. Only the results will be given below the proof of the recursion formulas, the orthogonality relations, and the normalization constants are obtained by methods analogous to those in Appendix IV ... 

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