Sury's Identity: Fibonacci & Lucas Sums

• Sury’s Identity is a fundamental relation that connects Fibonacci and Lucas numbers via weighted finite sums, highlighting symmetry in integer sequences.
• It is proven through both algebraic (Binet formula) and combinatorial (tiling interpretation) methods, underscoring its structural depth.
• Generalizations extend this identity to parameter families, shifted sums, and polynomial analogues with applications in number theory and algorithm analysis.

Sury’s identity is a foundational result in the combinatorics and algebra of integer sequences, relating the Fibonacci numbers, Lucas numbers, and weighted finite sums. Originating within the broader study of binomial and recursively defined sequences, it has attracted deep investigation due to its elegant form and extensive generalizations that connect combinatorics, number theory, and group actions. Its modern impact is evident in active research areas, including generalizations to Lucas sequences of arbitrary parameters, Dirichlet character sums, and polynomial analogues.

1. Definitions and Statement of Sury’s Identity

Let $\{F_n\}_{n\ge0}$ denote the Fibonacci numbers: $$F_0 = 0, \quad F_1 = 1, \quad F_{n+2} = F_{n+1} + F_n \;\;(n \ge 0),$$ and let $\{L_n\}_{n\ge0}$ denote the Lucas numbers: $$L_0 = 2, \quad L_1 = 1, \quad L_{n+2} = L_{n+1} + L_n \;\;(n \ge 0).$$ By induction, $L_n = F_{n-1} + F_{n+1}$ with the convention $F_{-1}=1$. Using the Binet forms,

$$F_n = \frac{\phi^n - \psi^n}{\phi - \psi}, \quad L_n = \phi^n + \psi^n, \quad \phi=\tfrac{1+\sqrt{5}}{2},\;\psi=\tfrac{1-\sqrt{5}}{2},$$

Sury’s identity asserts that for every $n \ge 0$,

$\sum_{k=0}^n 2^k\,L_k = 2^{\,n+1}\,F_{n+1}.$

This result, first appearing in binomial identity literature and refined by B. Sury (Martinjak, 2015), exhibits remarkable symmetry in the weighted interplay between Lucas and Fibonacci sequences.

2. Proofs of the Original Identity

Algebraic (Binet) Derivation

Using the explicit forms for $L_k$,

$$\sum_{k=0}^n 2^k L_k = \sum_{k=0}^n (2\phi)^k + \sum_{k=0}^n (2\psi)^k.$$

Summing the geometric series and substituting $2\phi-1 = \phi^2$, $2\psi-1 = \psi^2$, one obtains: $$\sum_{k=0}^n 2^k L_k = \frac{(2\phi)^{n+1} - 1}{\phi^2} + \frac{(2\psi)^{n+1} - 1}{\psi^2} = 2^{n+1} F_{n+1}.$$ This verification hinges on the arithmetic of the characteristic polynomial roots and the telescoping structure inherent in the Binet formula (Martinjak, 2015, 2312.02223, Bhatnagar, 2015).

Combinatorial/Bijective Derivation

Combinatorial proofs rest on tiling interpretations. $F_{n+1}$ counts the tilings of a $1 \times n$ board with squares and dominoes, while $L_k$ counts such tilings on a cyclic “bracelet” of length $k$. A weighted sum over all bracelet lengths, with $2^k$-coloring for squares and dominoes, doubles the total count of colored board-tilings, matching $2^{n+1} F_{n+1}$ via explicit bijection (Martinjak et al., 2015).

3. Classical and Modern Extensions

Direct Generalizations

Key one-parameter generalizations include

$$\sum_{k=0}^n m^k\left(L_k + (m-2) F_{k+1}\right) = m^{n+1} F_{n+1} \quad (m \ge 2).$$

This interpolation, with $m=2$ recapturing Sury’s identity, produces infinite families of identities parameterized by $m$ (Martinjak et al., 2015).

Another family: $$\sum_{k=0}^n (-1)^k\,m^{n-k} \left(L_{k+1} + (m-2)F_k\right) = (-1)^n F_{n+1},$$ illustrates the algebraic dual to the first summation identity.

Shifted and Weighted Sums

Polynomial and shifted variants have appeared: $$\sum_{j=0}^n 2^j\,L_{j+r} = 2^{n+1} F_{n+r+1} - F_r,$$

$$\sum_{j=0}^n 2^j\,F_{j+r} = 2^{n+1} L_{n+r+1} - L_r,$$

as well as convolution formulas involving arbitrary linear combinations of Fibonacci and Lucas numbers (2312.02223, Dresden et al., 14 Dec 2025).

Recurrence Analogue

Weighted sum identities naturally extend to sequences $I_n$ with $I_{n+2} = a_n I_{n+1} + b_n I_n$. Euler’s telescoping lemma provides the master identity framework: $$\sum_{k=1}^n \frac{t^{k-1}(b_{k-1}I_{k-1} + (t-1)I_{k+1})}{a_1 a_2\cdots a_{k-1}} = \frac{t^n}{a_1\cdots a_n} I_{n+1} - 1,$$ from which Sury’s identity is a specialization for Fibonacci-Lukas pairs and $t = 2$ (Bhatnagar, 2015, Dresden et al., 14 Dec 2025).

4. Connections to Menon-Type Identities

Sury generalized Menon's identity, a result from arithmetical group action, to higher-order gcd sums: $$\sum_{\substack{a \in (\mathbb Z / n\mathbb Z)^*}} \gcd(a-1, n) = \varphi(n) \tau(n),$$ and in the extension,

$$\sum_{\substack{a, b_1, ..., b_r}} \gcd(a-1, b_1, \ldots, b_r, n) = \varphi(n) \sigma_r(n).$$

Further generalizations admit Dirichlet character weights, yielding

$$\sum_{\substack{a \in (\mathbb Z / n\mathbb Z)^*,\, b_1, ..., b_r \in \mathbb Z/n\mathbb Z}} \gcd(a-1, b_1, ..., b_r, n) \chi(a) = \varphi(n) \sigma_r(n/d),$$

where $d$ is the conductor of $\chi$ (Li et al., 2018, Chen et al., 2018, Chattopadhyay et al., 2020).

These identities are proven via reduction to prime-power cases, group filtrations on units, and explicit multiplicativity through the Chinese remainder theorem.

5. Highlighted Further Extensions

A significant recent development is the unification of various Sury-type identities via determinant and convolution methods (Dresden et al., 14 Dec 2025). These yield:

• Arbitrary shift/convolution: Identities of the form

$$F_{n+1} = \frac{1}{L_k} \sum_{i=0}^n \left(-\frac{L_{k-1}}{L_k}\right)^{n-i} L_{i+k},$$

and analogously for more general second-order sequences.

• $j$-step Fibonacci/Lucas analogues: Identities for $F_{j n}$ and $L_{j n}$, further linking the finite weighted sums to composite-index sequences.
• Polynomial and order-lifting: Analogous formulas for Horadam sequences, Fibonacci and Chebyshev polynomials, and other linear recurrences (2312.02223, Dresden et al., 14 Dec 2025).

A summary of representative families appears below:

Formula Family	General Form	Reference
Sury’s identity	$\sum_{k=0}^n 2^k L_k = 2^{n+1} F_{n+1}$	(Martinjak, 2015, Bhatnagar, 2015)
Parameter family	$$\sum_{k=0}^n m^k [L_k + (m-2)F_{k+1}] = m^{n+1} F_{n+1}$$	(Martinjak et al., 2015)
Shifted sum	$\sum_{j=0}^n 2^j L_{j+r} = 2^{n+1} F_{n+r+1} - F_r$	(2312.02223)
Arbitrary convolution	$$F_{n+1} = \frac1{L_k} \sum_{i=0}^n \left(-\frac{L_{k-1}}{L_k}\right)^{n-i} L_{i+k}$$	(Dresden et al., 14 Dec 2025)

6. Applications, Context, and Open Problems

Sury’s identity and its extensions surface in analytic, combinatorial, and algebraic contexts:

• Combinatorics: Weighted tiling enumerations, ballot-path counts, and colored tiling models utilize these identities for exact formula derivations (Martinjak et al., 2015).
• Generating Functions: Fundamental in weighted generating function manipulations, with clear interpretations in terms of power series coefficients for Fibonacci/Lucas recursions.
• Recurrence Theory: Provides canonical examples for general finite linear recurrences and divisibility properties in integer sequences (e.g., $F_r \mid F_{mr}$) (2312.02223).
• Number Theory: Through Menon-Sury-type identities, deep connections arise to arithmetic functions, Dirichlet characters, Dedekind domains, and ideal-theoretic sums (Li et al., 2018, Chen et al., 2018, Chattopadhyay et al., 2020).
• Algorithmic Analysis: Recurrences of the Sury type appear in the runtime analysis of recursive algorithms and in enumeration problems.

Open directions include generalization to polynomials and higher-order recurrences, further combinatorial interpretation beyond one-dimensional tilings, and systematization of divisibility properties implied by these weighted sums (Dresden et al., 14 Dec 2025).

7. Summary and Perspective

Sury’s identity is a paradigmatic example of duality and symmetry among classical recurrences, bridging combinatorial enumeration, algebraic identities, and arithmetic group actions. Modern research has shown its natural place as the simplest instance in broad families of weighted and shifted sum relations for linear recurrences and has uncovered its deeper roles in algebraic and analytic number theory. Current trends aim to extend these connections to polynomial analogues, higher-order recurrences, and to structure theory for integer sequences via convolution and transform identities (Dresden et al., 14 Dec 2025, 2312.02223).

The study of Sury’s identity thus continues to inform a wide spectrum of mathematical and computational disciplines.