Golden tuple products (Q2883393)

Although not particularly long, the paper is however a bit intricate and the author wisely suggests reference links from \url{http://mathworld.wolfram.com/} by \textit{E. Weisstein} in order to clarify some of the necessary key-notions.NEWLINENEWLINENEWLINEIn this paper the triple product \(TP\) of \textit{Jacobi}, classically described by \textit{G. E. Andrews} [The theory of partitions. Cambridge: Cambridge University Press (1998; Zbl 0996.11002)], is introduced in the revisited identities by \textit{D. Foata} and \textit{G.-N. Han} [Sémin. Lothar. Comb. 42, B42o, 12 p. (1999; Zbl 0923.11143)], while the quintuple product \(QP\) of \textit{G. N. Watson} [Journ. L. M. S. 4, 39--48 (1929; JFM 55.0273.01)] is presented in the forms supplied by \textit{H. M. Farkas} and \textit{I. Kra} [Proc. Am. Math. Soc. 127, No. 3, 771--778 (1999; Zbl 0932.11029)] and by \textit{S.-Y. Kang} [J. Comb. Theory, Ser. A 78, No. 2, 313--318 (1997; Zbl 0944.11033)]. In all these multiple products the variables \(z\) and \(q\) are complex, with \(z\neq 0\) and \(|q|<1\).NEWLINENEWLINENEWLINESince the identities of the revisited \(TP\) and \(QP\) hold as formal power series in \(q\), in the paper these series expansions are balanced (i.e., divided by the complex variable \({\frac{z-1}{z}}\)), simplified through the \(q\)-Pochhammer symbol (commonly used in \(q\)-series notation), modified by substituting \(r=z+z^{-1}\) and \(p=z^2+z^{-2}=r^2-2\) and summed over the index \(k^{\ast}=|k|\) for \(k^{\ast}\geq 0\).NEWLINENEWLINENEWLINEThe author develops polynomials in \(q\) with coefficients \({\alpha_i (r)}\), \({\beta_i (r)}\), \({\gamma_i (r)}\), defined both by recursion and in terms of Chebyshev polynomials of the first and second kind, from which he obtains bijections of the Fibonacci and Lucas sequences when \(r=\pm 3\).NEWLINENEWLINENEWLINEAbout the expansion coefficients of the multiple products, the author points out that: they come from lacunary sequences; they can be generalized for each balanced multiple product by substituting an array of coefficients \({\gamma_i (r)}\) for the single variable \(r\); they have special connections with Fibonacci and Lucas numbers so that the ratios of their pairings at \(\pm r\) yield convergents to surds; they can be subject to an additional pairing of like powers (e.g., \(z^n\) and \(z^{-n}\)) implying the use of the standard Binet form analysis well illustrated by \textit{N. J. A. Sloane} and \textit{S. Plouffe} [The encyclopedia of integer sequences. Incl. 1 IBM/MS-DOS disk. (3.5). San Diego, CA: Academic Press (1995; Zbl 0845.11001)].NEWLINENEWLINEAs main results, the author establishes various new identities, many of which proved on the inductive basis derived by matching coefficients in Binet form, he generalizes the convergents of the square roots of numbers in terms of the variable \(r\) and he examines the balanced multiple product formulas not only with integer values of \(r\) (being so able to identify many known number sequences) but also with strictly imaginary values for such variable, thus finding new coefficient identities not exhaustively analyzed yet.NEWLINENEWLINEBeyond Fibonacci, Lucas, Pell and Pell-Lucas numbers, in the proof the author employs: Dellannoy and octagonal numbers, some methods for Fibonacci and related sequences expounded by \textit{R. C. Johnson} [``Fibonacci resources'', \url{http://www.dur.ac.uk/bob.johnson/fibonacci/}] and formulas involving Fibonacci and Lucas numbers such as the double angle formula \(F_{2n}=L_n F_n\) and several others chosen from the huge set provided by \textit{S. Vajda} [Fibonacci \& Lucas numbers, and the golden section. Theory and applications. Chichester: Ellis Horwood Ltd. etc. (1989; Zbl 0695.10001)] and further extended in this paper.