New Binomial Identities for Fibonacci, Lucas, and Generalized Fibonacci Sequences with Multiple Indices

Abstract: This paper presents new identities expressing the terms of Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices through powers of Lucas numbers and binomial coefficients. The obtained formulas rely on the application of symmetric polynomials (Waring's formulas) to the classical Binet's formula. Particular attention is given to the binomial expansion for the generalized Fibonacci sequence, which structurally combines two adjacent binomial coefficients from Pascal's triangle.

• The paper derives explicit binomial-polynomial formulas for F_nm and L_nm in terms of L_n, with parity controlled by the factor (-1)^{i(n+1)}.
• The paper extends the method to arbitrary generalized Fibonacci sequences, expressing G_nm through adjacent Pascal-triangle coefficients tied to G_n and G_0.
• The proofs combine Binet’s formula, symmetric-polynomial identities, Chebyshev structure, and d’Ocagne’s identity, offering exact formulas but no benchmark against fast-doubling algorithms.
• How do the binomial identities for F_nm and L_nm relate to Chebyshev polynomials and Lucas triangle coefficients?
• Can the generalized Fibonacci identity be proved combinatorially without using Binet’s formula or d’Ocagne’s identity?
• Do these multiple-index identities extend to negative indices, higher-order recurrences, or q-Fibonacci sequences?
• How might the formulas improve or compare with fast doubling and matrix exponentiation for computing large-index terms?
• Find recent papers about binomial identities for generalized Fibonacci sequences.

Overview

This paper establishes three explicit binomial identities expressing multiple-index terms of the Fibonacci sequence $F_{nm}$, the Lucas sequence $L_{nm}$, and the generalized Fibonacci sequence $G_{nm}$ as polynomials in the single Lucas number $L_n$ with binomial coefficients drawn from Pascal's triangle. The proofs are elementary and uniform in method: Binet's formula reduces each multiple-index term to a symmetric function of the $n$-th powers of the golden-ratio conjugates $\alpha$ and $\beta$, and Waring's formulas for power sums convert those symmetric functions into polynomials in the elementary symmetric polynomials $S = x + y = L_n$ and $P = xy = (\alpha\beta)^n = (-1)^n$.

Methodological framework

The paper rests on three classical results. First, Binet's formula gives $F_n = (\alpha^n - \beta^n)/\sqrt{5}$ and $L_{nm}$0, together with the relation $L_{nm}$1. Second, Waring's formulas express the two fundamental symmetric functions of two variables in terms of $L_{nm}$2 and $L_{nm}$3: the divided difference $L_{nm}$4 as an alternating sum over $L_{nm}$5, and the power sum $L_{nm}$6 as an alternating sum over $L_{nm}$7. These formulas are precisely the closed forms of Chebyshev polynomials of the second and first kinds, which the author highlights as the structural bridge between the recurrence sequences and symmetric-polynomial theory. Third, d'Ocagne's identity $L_{nm}$8 is needed only for the generalized case.

The Fibonacci and Lucas identities

Theorem on $L_{nm}$9: substituting $G_{nm}$0, $G_{nm}$1 into the divided-difference form of Waring's formula yields

$G_{nm}$2

The sign factor $G_{nm}$3 arises from combining the alternating sign of Waring's formula with the product term $G_{nm}$4. This identity expresses the ratio $G_{nm}$5 entirely as a polynomial in $G_{nm}$6, so no division or irrational arithmetic is required once $G_{nm}$7 is known — a point of direct algorithmic relevance.

Theorem on $G_{nm}$8: the power-sum form of Waring's formula gives

$G_{nm}$9

Here the coefficients $L_n$0 are the Lucas triangle coefficients, and the same sign factor $L_n$1 governs the parity structure inherited from $L_n$2. Both identities are direct consequences of the substitution and require no additional machinery; their correctness is fully determined by the two Waring identities, which the paper cites rather than re-proves.

The generalized Fibonacci identity

The main technical contribution is the identity for the generalized sequence $L_n$3 with arbitrary initial conditions $L_n$4. The proof exploits the representation $L_n$5 and eliminates $L_n$6 via $L_n$7, producing

$L_n$8

The $L_n$9-coefficient is exactly the Fibonacci ratio from the first theorem. The $n$0-coefficient requires the d'Ocagne step: $n$1, after which the Fibonacci theorem is applied with parameter $n$2 and the summation index is shifted by one ($n$3). The structural outcome is that the two sums use adjacent binomial coefficients from Pascal's triangle — $n$4 and $n$5 at matching values of $n$6 — with the $n$7 sum offset by one in its lower limit. This adjacency is presented by the author as the notable feature of the generalized formula, and it means both coefficients can be read off a single row-diagonal structure of Pascal's triangle.

One caveat in the derivation: the elimination of $n$8 involves division by $n$9, which is legitimate since $\alpha$0 guarantees $\alpha$1, but the intermediate expressions are rational in $\alpha$2 even though the final identity is polynomial in $\alpha$3 and integer-valued. The paper does not discuss whether the identity extends to negative $\alpha$4 or to the Lucas companion of the generalized sequence.

Limitations and open questions

The results are confined to the two-step recurrence with characteristic roots $\alpha$5; no extension to higher-order recurrences or to $\alpha$6-Fibonacci analogues is attempted. The identities are proven for positive integers $\alpha$7 only. The paper also does not quantify the computational advantage of the binomial form over fast doubling or matrix-power methods for evaluating $\alpha$8, so the claimed algorithmic applicability remains asserted rather than benchmarked. Whether the adjacent-coefficient structure of the generalized identity admits a direct combinatorial proof — bypassing the Binet/Waring route — is left unaddressed.

Conclusion

The paper provides short, rigorous proofs that $\alpha$9, $\beta$0, and $\beta$1 are polynomials in $\beta$2 with explicit binomial coefficients, unified through the application of Waring's formulas to Binet's formula. The generalized Fibonacci identity, with its paired adjacent binomial coefficients and single sign factor $\beta$3, is the paper's most distinctive result, connecting multiple-index evaluation of second-order recurrences to the combinatorics of Pascal's triangle and the Chebyshev polynomials.