Abstract
The ladder structure is briefly explained, and its essential characteristics described. Using Kummer's functional equations and a generalization of Roger'sL-function, it is proved that a) the transcendental part of the ladder retains its structure when the order is decreased; b) the logarithmic part also retains its structure; and c) as a special case of the latter, a cyclotomic equation for the variable, as previously observed empirically, can be shown to be satisfied. With some refinements, the order can also be increased beyond the fifth, to where there currently are no known relevant functional equations. Flow charts are given to show the structure of the order-augmentation process. The existence of a new category of such functional equations is conjectured, and some examples, for lower orders, are presented.
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Lewin, L. The order-independence of the polylogarithmic ladder structure—implications for a new category of functional equations. Aeq. Math. 30, 1–20 (1986). https://doi.org/10.1007/BF02189908
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DOI: https://doi.org/10.1007/BF02189908