OON THIRD-ORDER PELL POLYNOMIALS
HELMUT PRODINGERA
BSTRACT . Binet formulae for three versions of third-order Pell polynomials are derived.
1. A B
INET FORMULA FOR THIRD - ORDER P ELL POLYNOMIALS
Mahon and Horadam [ ] study the recursion for r n = r n ( x ) r n = x r n − + r n − , n ≥ r = r = r = x .The characteristic equation of the recursion is X − x X − = [ ] the authors try to attack it with Cardano’s formula (and without using computers).Based on our recently acquired experience with ternary recursions, see, e. g., [ ] , and using Maple , we are able to get stronger and quite satisfying results.First, substitute X = x / Y and the equation is now Y + x Y − x = x = − / z it is zY − Y + = z = ( − t ) ( + t ) ,the equation has beautiful roots: v : = − t , v : = + t + (cid:112) ( + t )( − t ) ( t − ) , v : = + t − (cid:112) ( + t )( − t ) ( t − ) .Note that v + v = / ( t − ) and v v = / ( t − ) . Setting r n = x n − (cid:128) Av − n + B v − n + C v − n (cid:138) . Mathematics Subject Classification.
Key words and phrases.
Pell polynomials, Binet formula, Girard-Waring formula, third-order recursion. a r X i v : . [ m a t h . N T ] D ec HELMUT PRODINGER we can now compute the coefficients A , B , C from the initial values, with the result A = − + t , B = ( + t ) − (cid:112) ( + t )( − t )( − t ) ( + t ) , C = ( + t ) + (cid:112) ( + t )( − t )( − t ) ( + t ) .Even more appealing are the reciprocal roots: w = v − = − tw = v − = + t − (cid:112) ( + t )( − t ) w = v − = + t + (cid:112) ( + t )( − t ) r n = x n − (cid:128) Aw n + Bw n + C w n (cid:138) .As an example, r = x + x + x + x + x + x It starts with the highest power x , and goes down from there by powers of x − , as predicted.2. T HE COEFFICIENTS
First, we want to show, that as in the Binet formula for Fibonacci numbers, the square rootis superficial and cancels out. To see this, we compute ( + t ) r n x n − = − ( − t ) n + (cid:149) − (cid:112) ( + t )( − t )( − t ) (cid:152)(cid:129) + t − (cid:112) ( + t )( − t ) (cid:139) n + (cid:149) + (cid:112) ( + t )( − t )( − t ) (cid:152)(cid:129) + t + (cid:112) ( + t )( − t ) (cid:139) n .The non-trivial part (lines 2 and 3), multiplied by 2 n + and the abbreviation W = (cid:112) ( + t )( − t ) is Ξ : = (cid:149) − W − t (cid:152)(cid:0) + t − W (cid:1) n + (cid:149) + W − t (cid:152)(cid:0) + t + W (cid:1) n N THIRD-ORDER PELL POLYNOMIALS 3 = (cid:88) k (cid:129) n k (cid:139) W k ( + t ) n − k + (cid:88) k (cid:129) n k + (cid:139) W k + ( + t ) n − − k W − t = (cid:88) k (cid:129) n k (cid:139) ( − t ) k ( + t ) n − k + (cid:88) k (cid:129) n k + (cid:139) ( − t ) k ( + t ) n − k ,and no more square roots are present.Now we want to show how to compute various series expansions. We start with z = ( − t ) ( + t ) .So, z is given in terms of t , but we want the expansion of t in terms of z . This can be seenin the context of the Lagrange inversion or the Lagrange-Bürmann formula, [ ] . We will usecontour integration to get the inverted series. We need another substition: t = u − = ⇒ z = u ( u − ) .The advantage is that z → ⇔ u →
0; we restrict ourselves to | u | < . FurthermoreF IGURE z as a function of u . If | u | < the function can be inverted. dzdu = ( u − )( u − ) . We are going to show that u = (cid:88) n ≥ n (cid:129) n − n − (cid:139) n − z n .The usual convergence test shows that this series converges for | z | < ≈ [ z n ] u = π i (cid:73) dzz n + u = π i (cid:73) du ( u − )( u − ) u n + ( u − ) n + u = [ u n − ] u − ( u − ) n + = n + [ u n − ] − u ( − u ) n + . HELMUT PRODINGER
Reading off this coefficient and simplifying leads to the result. Now we will do a similarcomputation to show that − ( − t ) n + t = (cid:88) (cid:96) ≥ n − (cid:96) − (cid:129) (cid:96) − n (cid:96) (cid:139) z (cid:96) ,which is the first term of the Binet formula. First, − ( − t ) n + t = − ( − u ) n u − − [ z (cid:96) ] ( − u ) n u − = − π i (cid:73) dzz (cid:96) + ( − u ) n u − = − π i (cid:73) du ( u − )( u − ) u (cid:96) + ( u − ) (cid:96) + ( − u ) n u − = − ( − ) n π i (cid:73) duu (cid:96) + ( u − ) (cid:96) + − n = [ u (cid:96) ] ( − u ) (cid:96) + − n = (cid:96) + − n (cid:129) (cid:96) − n (cid:96) (cid:139) .If only this first term of the Binet formula would be used, we would have derived that r n ( x ) = (cid:88) (cid:96) ≥ (cid:129) n − − (cid:96)(cid:96) (cid:139) ( x ) n − − (cid:96) .However, r n ( x ) isn’t an infinite series, it is just a polynomial. This means that the secondand third term in the Binet formula kill off the infinite rest of the series. This can be seen forinstance, by finding the expansions of w n + w n and w n − w n w − w using the Girard-Waring formula,viz. [ ] .Instead of showing this cancellation directly, we prove that r n ( x ) = (cid:88) ≤ (cid:96) ≤ ( n − ) / (cid:129) n − − (cid:96)(cid:96) (cid:139) ( x ) n − − (cid:96) by simple induction. The initial conditions match, and then: r n = x r n − + r n − = x (cid:88) ≤ (cid:96) ≤ ( n − ) / (cid:129) n − − (cid:96)(cid:96) (cid:139) ( x ) n − − (cid:96) + (cid:88) ≤ (cid:96) ≤ ( n − ) / (cid:129) n − − (cid:96)(cid:96) (cid:139) ( x ) n − − (cid:96) = (cid:88) ≤ (cid:96) ≤ ( n − ) / (cid:129) n − − (cid:96)(cid:96) (cid:139) ( x ) n − − (cid:96) + (cid:88) ≤ (cid:96) ≤ ( n − ) / (cid:129) n − − (cid:96)(cid:96) − (cid:139) ( x ) n − − (cid:96) = ( x ) n − + (cid:88) ≤ (cid:96) ≤ ( n − ) / (cid:129) n − − (cid:96)(cid:96) (cid:139) ( x ) n − − (cid:96) + [[ | ( n − )]]= (cid:88) ≤ (cid:96) ≤ ( n − ) / (cid:129) n − − (cid:96)(cid:96) (cid:139) ( x ) n − − (cid:96) , N THIRD-ORDER PELL POLYNOMIALS 5 as predicted. 3. T
HE SECOND VERSION OF THE THIRD - ORDER P ELL POLYNOMIALS
We consider the second version proposed by Mahon and Horadam, but only indicate whatchanges. These polynomials are given by s n = x s n − + s n − , n ≥ r = r = r = x .Since the recursion is the same, only the initial values change, and they are A = t ( + t )( t − ) B = t ( + t )( t − ) + t − t − ( t − )( t + )( t − ) C = t ( + t )( t − ) − t − t − ( t − )( t + )( t − ) The Binet formula is then s n = x n − (cid:0) Aw n + Bw n + C w n (cid:1) .There is also an explicit formula, which works for n ≥ s n ( x ) = (cid:88) ≤ (cid:96) ≤ ( n − ) / ( x ) n − − (cid:96) n − (cid:96) − n − (cid:96) − (cid:129) n − (cid:96) − (cid:96) (cid:139) As before, this could be guessed from the first term in the Binet formula and proved byinduction. 4. T
HE THIRD VERSION OF THE THIRD - ORDER P ELL POLYNOMIALS
These polynomials are given by σ n = x σ n − + σ n − , n ≥ σ = σ = x , σ = x .This time the Binet formula is very simple: σ n = x n ( w n + w n + w n ) .There is an explicit formula, valid for n ≥ σ n = (cid:88) ≤ (cid:96) ≤ n / nn − (cid:96) (cid:129) n − (cid:96)(cid:96) (cid:139) ( x ) n − (cid:96) HELMUT PRODINGER R EFERENCES [ ] H. Gould. The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences.
TheFibonacci Quarterly , 37:135–140, 1999. [ ] P. Henrici.
Applied Complex Analysis Volume 1 . John Wiley, 1988. [ ] Br. J. M. Mahon and A. F. Horadam. Third-order diagonal functions of Pell polynomials.
The FibonacciQuarterly , 28:3–10, 1990. [ ] Helmut Prodinger. On some problems about ternary trees — a linear algebra approach.
The Rocky MountainsJournal of Mathematics , 2021.H
ELMUT P RODINGER , D
EPARTMENT OF M ATHEMATICAL S CIENCES , S
TELLENBOSCH U NIVERSITY , 7602 S
TELLEN - BOSCH , S
OUTH A FRICA
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