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Number Theory

Sums of Squares: Methods for Proving Identity Families

Profile image of Russell HendelRussell Hendel

2021, arXiv (Cornell University)

https://doi.org/10.48550/ARXIV.2103.16756
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11 pages

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Abstract

This paper presents both a result and a method. The result presents a closed formula for the sum of the first m`1, m ě 0, squares of the sequence F pkq where each member is the sum of the previous k members and with initial conditions of k´1 zeroes followed by a 1. The generalized result includes the known result of sums of squares of the Fibonacci numbers and recent results of Ohtsuka-Jakubczyk, Howard-Cooper, Schumacher, and Prodinger-Selkirk for the cases k " 2, 3, 4, 5, 6. The paper contributes a closed formula for coefficients for all k. To prove the result, the paper introduces a new method, the algebraic verification method, which reduces proof of an identity to verification of the equality of finitely many pairs of finite-degree polynomials, possibly in several variables. Additionally, the paper provides a visual aid, labeled index squares, for complicated proofs. Several other papers proving families of identities are examined; it is suggested that the collection of the uniform proof methods used in these papers could possibly produce a new trend in stating and proving identities.

Figures (11)
Table 1: Coefficients, N;,; of G2 i<k-—1for2 < k < 6, based on (3)-(7). m+i? O< This table shows clear linear patterns in each column which naturally leads to a conjecture on the functional form of the N;;. The cases 2 = 0 andi > 1 must be treated separately. For 2 = 0, The obvious approach to finding the pattern in the N;,;, creating a table whose rows are labeled with k and whose columns are labeled with F' Ah) Ee ; does not immediately yield results. The correct procedure that facilitates dentification of patterns in the coefficients is to consider the diagonal elements (that is, the case i = 7) and the non- liagonal elements separately and in different ways. The table of coefficients for the diagonal elements is presented in [able 1.
Table 1: Coefficients, N;,; of G2 i<k-—1for2 < k < 6, based on (3)-(7). m+i? O< This table shows clear linear patterns in each column which naturally leads to a conjecture on the functional form of the N;;. The cases 2 = 0 andi > 1 must be treated separately. For 2 = 0, The obvious approach to finding the pattern in the N;,;, creating a table whose rows are labeled with k and whose columns are labeled with F' Ah) Ee ; does not immediately yield results. The correct procedure that facilitates dentification of patterns in the coefficients is to consider the diagonal elements (that is, the case i = 7) and the non- liagonal elements separately and in different ways. The table of coefficients for the diagonal elements is presented in [able 1.
Table 2: Coefficients N;,;,0 <i <k—2,1+1<j<k—1, for the case k = 6, based on (7). Table 2, like Table 1, exhibits linear patterns which motivate the following conjecture on the functional form of ian aiaanes — ied” ana, ie Sia” Dl The non-diagonal coefficients N;,;,2 # j, naturally form a 3-dimensional solid rather than a triangle; more specif- ically for each k, the coefficients N;,;,7 = 7+ 1, form a triangle. For k = 6 this triangle is shown in Table 2. For 2 < k < 5, the corresponding triangles are easy to construct; (they may also be found (albeit crossed out) at [9, Sequence A342955, history, item #43]).
Table 2: Coefficients N;,;,0 <i <k—2,1+1<j<k—1, for the case k = 6, based on (7). Table 2, like Table 1, exhibits linear patterns which motivate the following conjecture on the functional form of ian aiaanes — ied” ana, ie Sia” Dl The non-diagonal coefficients N;,;,2 # j, naturally form a 3-dimensional solid rather than a triangle; more specif- ically for each k, the coefficients N;,;,7 = 7+ 1, form a triangle. For k = 6 this triangle is shown in Table 2. For 2 < k < 5, the corresponding triangles are easy to construct; (they may also be found (albeit crossed out) at [9, Sequence A342955, history, item #43]).
Table 3: The seven groups of Gim+4iGm+j The following examples are illustrative of the use of the index-pair sets and labeled index squares. Example 4.1. 3), Ni+1,541Gm+iGm+j = No,oG?,_1. In this example, by Table 3, A = {i = —1,j = —1}. Hence Ni41541Gm+iGm+; = No0G2,_, as required.
Table 3: The seven groups of Gim+4iGm+j The following examples are illustrative of the use of the index-pair sets and labeled index squares. Example 4.1. 3), Ni+1,541Gm+iGm+j = No,oG?,_1. In this example, by Table 3, A = {i = —1,j = —1}. Hence Ni41541Gm+iGm+; = No0G2,_, as required.
Table 4: Summand-I = Vio<i<x—2 Ni,jG@m+iGm+j = Up CFG NijG@m+iGme4; t<j<k—2 This section implements Step vi of the proof, analyzing the five summands of the last section according to the index pairs over which they are summed. For the convenience of the reader, the five summands are repeated in the header of each table. The labeled index squares provide visual aids and review the definition of the seven index sets. Each cell of these labeled index squares contain i) the index letter for that pair of indices and ii) the linear combination of N;,; for that cell. These two items are separated by a comma. The index-squares should be fairly straightforward to read and check.
Table 4: Summand-I = Vio<i<x—2 Ni,jG@m+iGm+j = Up CFG NijG@m+iGme4; t<j<k—2 This section implements Step vi of the proof, analyzing the five summands of the last section according to the index pairs over which they are summed. For the convenience of the reader, the five summands are repeated in the header of each table. The labeled index squares provide visual aids and review the definition of the seven index sets. Each cell of these labeled index squares contain i) the index letter for that pair of indices and ii) the linear combination of N;,; for that cell. These two items are separated by a comma. The index-squares should be fairly straightforward to read and check.
" 2 2 Table 7: Summand-IV = — > -1<ich—2 Ni+i 41 Gig = —D4,B,c Ni+it iti Giri
" 2 2 Table 7: Summand-IV = — > -1<ich—2 Ni+i 41 Gig = —D4,B,c Ni+it iti Giri
Table 8: Summand-V = — >)-1<ij<p—3 Ni+1,341Gm+iGm4+j = — DD,E,F.G Ni41,j41Gm4iGm+j i<j<k—2 .
Table 8: Summand-V = — >)-1<ij<p—3 Ni+1,341Gm+iGm4+j = — DD,E,F.G Ni41,j41Gm4iGm+j i<j<k—2 .
Table 9: Coefficient of Gm+4iGm-+j in Summand-I through Summand-V as given in (15)-(16). Next, we implement Step ix. Using (8)-(11) we evaluate each entry in Table 9 as a polynomial in the variables i,j,k.
Table 9: Coefficient of Gm+4iGm-+j in Summand-I through Summand-V as given in (15)-(16). Next, we implement Step ix. Using (8)-(11) we evaluate each entry in Table 9 as a polynomial in the variables i,j,k.
Table 10: Coefficient of Gm +;Gm-+ j in equations Summand-I through Summand-V. The sum column, accomplished through paper and pencil summation of polynomials of each row, implements Step x of the proof outline and calculates the Nj, ; As can be seen from (14), the sum column confirms the algebraic verification of polynomial equality of the sides of (14) since, by (8), N’:,; is equal to D;, fori = 0 = 7 and 0 otherwise as required.
Table 10: Coefficient of Gm +;Gm-+ j in equations Summand-I through Summand-V. The sum column, accomplished through paper and pencil summation of polynomials of each row, implements Step x of the proof outline and calculates the Nj, ; As can be seen from (14), the sum column confirms the algebraic verification of polynomial equality of the sides of (14) since, by (8), N’:,; is equal to D;, fori = 0 = 7 and 0 otherwise as required.

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References (13)

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  2. Russell Jay Hendel; "Review of Subsequences of Fibonacci and Lucas polynomials with geometric subscripts, Fibonacci Quart., 50, 27-35," Mathematical Reviews, #2892007, (2012).
  3. Russell Jay Hendel, "Recursive Triangles Appearing Embedded in Recursive Families," Fibonacci Quarterly, 58, (2020), 135-144.
  4. Russell Jay Hendel, " A Method for Uniformly Proving a Family of Identities," Fibonacci Quarterly, To ap- pear,2022.
  5. Fred T. Howard and Curtis Cooper, "Some Identities for r-Fibonacci Numbers," The Fibonacci Quarterly, 49(3), 231-242, (2011).
  6. T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.
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  13. Raphael Schumacher, "How to Sum the Squares of the Tetranacci Numbers and the Fibonacci m-STEP Num- bers," Fibonacci Quarterly, 57(2), 168-175, (2019).

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