Summary
The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu p +u q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base θ, where θ3 + θ = 1, are developed through the sixth order.
Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu 5+u 3 = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.
The equationu 6m+1 +u 6r−1 = 1 is discussed and some valid results deduced. This equation is divisible byu 2 −u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.
New functional equations to give the supernumary θ-ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.
A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z m(1−z)r(1 +z)s are not capable of extension to the sixth order.
There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.
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References
Lewin, L.,The dilogarithm in algebraic fields. J. Austral. Math. Soc. A33 (1982), 302–330.
Lewin, L.,The inner structure of the dilogarithm in algebraic fields. J. Number Theory19 (1984), 345–373.
Abouzahra, M. andLewin, L.,The polylogarithm in algebraic number fields. J. Number Theory21 (1985), 214–244.
Lewin, L.,The order-independence of the polylogarithmic ladder structure — implications for a new category of functional equations. Aequationes Math.30 (1986), 1–20.
Lewin, L. andRost, E.,Polylogarithmic functional equations: a new category of results developed with the help of computer algebra (MACSYMA). Aequationes Math.31 (1986), 223–242.
Abouzahra, M. andLewin, L.,The polylogarithm in the field of two irreducible quintics. Aequationes Math.31 (1986), 315–321.
Abouzahra, M., Lewin, L. andHongnian, X.,Polylogarithms in the field of omega (a root of a given cubic): functional equations and ladders. Aequationes Math.33 (1987), 23–45. Addendum in35 (1988), 304.
Wechsung, G.,Über die Unmöglichkeit der Vorkommens von Funktionalgleichungen gewisser Struktur für Polylogarithmen. Aequationes Math.5 (1970), 54–62.
Browkin, J., personal communication.
Euler, L.,Institutiones Calculi Integralis. Bd.1 (1768), pp. 110–113. (see also [12]).
Landen, J.,Mathematical Memoirs. Vol. 1, 1780, p. 112 (see also [12]).
Lewin, L.,Polylogarithms and associated functions. Elsevier/North-Holland, New York, 1981.
Coxeter, H. S. M.,The functions of Schläfli and Lobatschefsky. Quart. J. Math. Oxford Ser.6 (1935), 13–29.
Phillips, M. J. andWhitehouse, D. J.,Two-dimensional discrete properties of random surfaces. Phil. Trans. Roy. Soc. London Ser. A305 (1982), 441–468.
Rogers, L. J., On function sum theorems connected with the series\(\sum _{n = 1}^\infty {{x^2 } \mathord{\left/ {\vphantom {{x^2 } {n^2 }}} \right. \kern-\nulldelimiterspace} {n^2 }}\). Proc. London Math Soc. (2)4 (1907), 169–189.
Loxton, J. H.,Special values of the dilogarithm function. Acta Arithmetica43 (1984), 155–166.
See equations (48) and (49) of Reference [2].
Browkin, J., private correspondence.Conjecture on the dilogarithm. To be published inK-Theory.
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Lewin, L., Abouzahra, M. Supernumary polylogarithmic ladders and related functional equations. Aeq. Math. 39, 210–253 (1990). https://doi.org/10.1007/BF01833152
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DOI: https://doi.org/10.1007/BF01833152