simpler Chebyshev inverse relations The so-termed simpler cases of the Chebyshev classes of inverse relations in the subsection below are given in the next table. The formulas in the table are simplified somewhat by the following identities: Additionally the inversion relations given in the table also hold when in any given relation. Chebyshev classes of inverse relations The terms, and , in the inversion formulas of the form for non-zero integers forming several special cases of Chebyshev classes of inverse relations are given in the next table. Additionally, these inversion relations also hold when for some or when the sign factor of is shifted from the terms to the terms . The formulas given in the previous table are simplified somewhat by the identities The simpler Legendre inverse relations Legendre–Chebyshev classes of inverse relations The Legendre–Chebyshev classes of inverse relations correspond to inversion relations of the form where the terms, and , implicitly depend on some fixed non-zero . In general, given a class of Chebyshev inverse pairs of the form if a prime, the substitution of , , and (possibly replacing ) leads to a Legendre–Chebyshev pair of the form Similarly, if the positive integer is composite, we can derive inversion pairs of the form The next table summarizes several generalized classes of Legendre–Chebyshev inverse relations for some non-zero integer . Abel inverse relations Abel inverse relations correspond to Abel inverse pairs of the form where the terms, and , may implicitly vary with some indeterminate summation parameter . These relations also still hold if the binomial coefficient substitution of is performed for some non-negative integer . The next table summarizes several notable forms of these Abel inverse relations. Inverse relations derived from ordinary generating functions If we let the convolved Fibonacci numbers, , be defined by we have the next table of inverse relations which are obtained from properties of ordinary sequence generating functions proved as in section 3.3 of Riordan's book. Note that relations 3, 4, 5, and 6 in the table may be transformed according to the substitutions and for some fixed non-zero integer . Inverse relations derived from exponential generating functions Let and denote the Bernoulli numbers and Euler numbers, respectively, and suppose that the sequences, , , and are defined by the following exponential generating functions: The next table summarizes several notable cases of inversion relations obtained from exponential generating functions in section 3.4