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The Jacobi Polynomial and the Perrin Number

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The Jacobi Polynomial and the Perrin Number The Jacobi Polynomial and the Perrin Number The Jacobi polynomial can also be used for the determination of the nth Perrin number. Mathematica is used for the calculations. As in Chapter 15 let, A0 = ((N1 − 3))⁄2 and B0 = 1 퐴 = RecurrenceTable[{푎[푛 + 1] == 푎[푛] − 3, 푎[0] == A0}, 푎, {푛, 0, Floor[A0⁄3]}] 퐵 = RecurrenceTable[{푎[푛 + 1] == 푎[푛] + 2, 푎[0] == B0}, 푎, {푛, 0, Floor[A0⁄3]}] Then the hypergeometric function [Eqn15.1.20] can be transformed as Floor[A0⁄3]+1 푆표 = ∑ (1⁄(퐴[[푛]] + 퐵[[푛]])) ∗ JacobiP[퐴[[푛]],0, (−(퐴[[푛]] + 퐵[[푛]]) − 1), −1] [P0] 푛=1 Or equivalently, Floor[A0⁄3]+1 푆표 = ∑ (−1)푛(1⁄(퐴[[푛]] + 퐵[[푛]])) ∗ JacobiP[퐴[[푛]], (−(퐴[[푛]] + 퐵[[푛]]) − 1), 0, 1] [P0-1] 푛=1 where So is the Sigma orbit and the Perrin number PN is 푃푁 = Round[N1 ∗ So] The equation is accurate for all numbers >12 (6 and 12 give one less than PN). For prime numbers let So be expressed as the Jacobi polynomial at x = 3. Then we can write, Floor[A0⁄3]+1 푆표 = ∑ (−1)푛(1⁄(퐴[[푛]] + 퐵[[푛]])) ∗ JacobiP[퐴[[푛]], (−(퐴[[푛]] + 퐵[[푛]]) − 1), 퐵[[푛]], 3] [P1] 푛=1 From [Eqs 4 and 19] So can be written as a sum of Jacobi polynomials; JacobiP[퐴[[푛]], (−(퐴[[푛]] + 퐵[[푛]]) − 1), 0,3] which can be reduced to [P2] JacobiP[퐴[[푛]], ((퐵[[푛]]) + 1) − (2 + 푥 ∗ 3),0,3] = JacobiP[퐴[[푛]], (−(퐴[[푛]] + 퐵[[푛]]) − 1), 0,3] Where x is an integer described below. I find that the value of original Jacobi polynomial in [P1] [P3] JacobiP[퐴[[푛]], (−(퐴[[푛]] + 퐵[[푛]]) − 1), 퐵[[푛]], 3] = Sn(i) where Sn(i) is the ith entry of the cycle index polynomial of the symmetry group Sn. [P4a,b] Sn = 푆퐴[[푛]] and I = 퐴[[푛]] + 1 + ((퐵[[푛]]) + 1) − (2 + 푥 ∗ 3) Example; For N = 47 the terms 퐴[[5]] = 10 푎푛푑 퐵[5] = 9 then Floor[A0⁄3] + 1 − 5 = 3 = 푥 ((퐵[[5]]) + 1) − (2 + 푥 ∗ 3) = 10 − 11 = −1 푖 = 퐴[[5]] + 1 + ((퐵[[5]]) + 1) − (2 + 3 ∗ 3) = 10 [P5] JacobiP[10, −20,9,3] = 푆10(10) = 92376 The expression for the cycle index polynomial for any symmetry group Sm given in Mathematica can also be written as [P6] |JacobiP[푚, −(푚 + 푖 − 1) − 1, 푖 − 1, 푥]| = |Binomial[m+i-1,m]| The values of i are integers and are limiting values as x approaches 1. All sigma orbits of prime numbers are represented by the symmetry groups represented in 퐴[[푛]] For the primes 23,29,41,47,59,71,83 and 89 the entries from S10 are respectively, 푆10(2), 푆10(4) , 푆10(8), 푆10(10) , 푆10(12), 푆10(14) , 푆10(18), 푆10(22), 푆10(24) Equation [P1] demonstrates another combinatoric role for the Jacobi Polynomial. It provides a direct calculation of the sigma orbit and Perrin number for any prime; and can be used to predict the value of the cycle index polynomial for symmetry groups. The limiting values of the Jacobi polynomial [P6] were found by in attempts to express the Jacobi polynomial in reduced forms. The analysis suggests that any prime is associated with the sigma orbit as a series of symmetry group entries that cannot be duplicated by composite numbers or Perrin pseudoprimes. Complex infinities are found when the Jacobi Polynomial in [P3] is used with composite numbers. These infinities are not found using [PO] or [PO-1]. RT 10/2/2018 .
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