Algebraic relations for reciprocal sums of Fibonacci numbers

Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values

“…Substituting the expansion z −2 + ∞ j=0 c j (k)z 2j of ns 2 (k, z) into these equations, we can determine the coefficients c j (k) (see [4,Lemma 2] and the recursive relation in it). The relation ns 2 (1, z) = coth 2 z immediately follows from the fact that u = ns 2 (1, z) satisfies the equation (u ) 2 = 4u(u − 1) 2 admitting the general solution coth 2 (z − z 0 ).…”

“…where Table 1(i)], [4]). The series A 2j+1 (β(k) 2 ) are generated by Tables 1(i)], [9], [11, p. 535]), namely…”

“…In [4], for ζ F (s) := ∞ n=1 F −s n , we proved that the values ζ F (2), ζ F (4), ζ F (6) are algebraically independent, and that for any integer s ≥ 4 ζ F (2s) − r s ζ F (4) ∈ Q(ζ F (2), ζ F (6)) with some r s ∈ Q; for example with u := ζ F (2), and v := ζ F (6).…”

“…For {h 2s } s∈N (respectively, {g * 2s−1 } s∈N ) algebraic relations were discussed in [6] (respectively, [5]). These sums may also be regarded as functions of the modulus k of Jacobian elliptic functions (see Sections 4 and 5, also [4], [5], [6]). …”

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

“…3). In the preceding paper [2] we proved that the numbers ζ F (2), ζ F (4), ζ F (6) are algebraically independent and for any integer s ≥ 4 ζ F (2s) − 5 s−2 r s ζ F (4) ∈ Q(u, v), u := ζ F (2), v := ζ F (6) with some r s ∈ Q (r s = 0 if and only if s is odd), where the rational function of u and v is explicit; for example,…”