[Paper Review] Convolution identities for Tetranacci numbers

Rusen Li|arXiv (Cornell University)|Sep 17, 2016

Advanced Mathematical Identities7 references3 citations

TL;DR

This paper establishes convolution identities for Tetranacci numbers using generating functions and exponential generating functions, deriving closed-form expressions without and with binomial coefficients. It introduces novel identities involving products of Binet-type coefficients and provides recurrence-based constructions for generalized Tetranacci-type sequences.

ABSTRACT

We give convolution identities without binomial coefficients for Tetranacci numbers and convolution identities with binomial coefficients for Tetranacci and Tetranacci-type numbers.

Motivation & Objective

• To derive convolution identities for Tetranacci numbers without binomial coefficients using ordinary generating functions.
• To develop convolution identities involving binomial coefficients for Tetranacci and Tetranacci-type numbers via exponential generating functions.
• To generalize Tetranacci sequences with arbitrary initial conditions and establish recurrence relations for their generating functions.
• To express products of Binet coefficients as exponential generating functions of new Tetranacci-type sequences.
• To provide a systematic method for constructing higher-order convolution identities through algebraic manipulation of generating functions.

Proposed method

• Utilizes the ordinary generating function $ T(x) = rac{x}{1 - x - x^2 - x^3 - x^4} $ to derive convolution identities via coefficient comparison.
• Applies the derivative $ T'(x) $ and algebraic manipulation to relate $ T(x)^2 $ to $ x^2 T'(x) $, yielding identities for $ n o ext{convolution sum} $.
• Employs the exponential generating function $ t(x) = rac{1}{563} rac{d}{dx} ext{some function} $ to express identities with binomial coefficients.
• Introduces modified Tetranacci numbers $ T_n^{(s_0,s_1,s_2,s_3)} $ with arbitrary initial values to generalize the recurrence structure.
• Uses partial fraction decomposition and root analysis of the characteristic equation $ x^4 - x^3 - x^2 - x - 1 = 0 $ to derive Binet-type formulas.
• Applies multinomial expansions and sign-adjusted sums over indices $ j,k $ to express coefficients in the generating function expansion.

Experimental results

Research questions

• RQ1 What convolution identities without binomial coefficients can be derived for Tetranacci numbers using ordinary generating functions?
• RQ2 How can convolution identities with binomial coefficients be constructed for Tetranacci and Tetranacci-type numbers using exponential generating functions?
• RQ3 Can products of Binet coefficients $ c_i c_j $ be expressed as generating functions of new Tetranacci-type sequences?
• RQ4 What recurrence relations govern the initial values and normalization constants of generalized Tetranacci sequences arising from such identities?
• RQ5 How can higher-order convolution identities be systematically generated using algebraic manipulation of generating function components?

Key findings

• For $ n o 4 $, the identity $ extstyleinom{n-4}{k=0} T_k (T_{n-k} + T_{n-k-2} + 2T_{n-k-3} + 3T_{n-k-4}) = (n-2)T_{n-1} - T_{n-2} - 3T_{n-3} $ holds.
• The convolution sum $ extstyleinom{n}{k=0} T_k T_{n-k} $ equals $ extstyleinom{n-2}{l=0} (l+1)T_{l+1} D $, where $ D $ is a complex sum over $ j,k $ with binomial and sign terms.
• The generating function $ extstylerac{1}{563} ext{sum} T_k^{(146,416,581,1080)} rac{x^k}{k!} $ equals a linear combination of $ e^{ heta x} $ terms with coefficients $ c_i c_j + c_j c_k + c_k c_i $.
• For $ n o 1 $, the sum $ (c_2c_3 + c_3c_4 + c_4c_2)e^{ heta x} + ext{cyclic terms} $ generates a Tetranacci-type sequence with initial values $ (146,416,581,1080) $.
• A recurrence for $ A_3^{(n)} $ and initial values $ s_{3,i}^{(n)} $ is derived such that $ (c_i c_j + ext{cyclic})^n e^{ heta x} $ generates $ rac{1}{A_3^{(n)}} ext{sum} T_{3,k}^{(s_{3,0}^{(n)}, ext{...})} rac{x^k}{k!} $.
• Normalization constants $ A_3^{(n)} $ are updated via a rational function of previous $ s_{3,i}^{(n)} $, ensuring positivity of the resulting sequence for large $ k $