SOME LINEAR RECURRENCES WITH CONSTANT COEFFICIENTS

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1 SOME LINEAR RECURRENCES WITH CONSTANT COEFFICIENTS RICHARD J MATHAR Abstract Linear homogeneous s-order recurrences with constant coefficients of the form a(n) d a(n ) + d 2 a(n 2) + + d sa(n s), n n r, have generating functions A(x), A(x) X P n a(i)x i a(n)xn P s i d ix i P n r i a(n)x n P s i0 i d ix i, rational functions in x, where a(0), a() up to a(r), s r +, are a set of the first few coefficients of the Taylor series which are set up independently Introduction Aim and Notation We consider the (ordinary) generating functions A(x) of sequences a(n) of numbers indexed by n 0,,, which are the expansion coefficients of the Taylor series [5 () A(x) a(i)x i i0 The generating functions are shown normalized in the sense that the first power of the Taylor expansion is the constant one; other offsets are essentially obtained by multiplication of the generating function with powers of x to shift the index up by arbitrary amounts Sequences with period length p after some optional non-periodic lower indices, a(n) a(n p), or sequences with period length p and a half-period symmetry (odd symmetry in the spea of Fourier Transforms), a(n) a(n p/2), are just special cases of these recurrences, with values d i equal to one or zero Generating functions A(x) with recurrences of constant coefficients and the restricted formats for inhomogeneities considered here are rational functions of x Decompositions in partial fractions my help to write down the generating functions as sums of two or more other generating functions, which in turn means that a sequence may be a term-by-term sum of more primitive sequences that may be investigated by some ind of reverse engineering (In these cases, PURRS [2 may propose closed-form expressions for a(n)) Sections 2 and 3 are explicit evaluations of the formula in the abstract for some simple cases; their limiting ratio are obtained from the characteristic function [0, 9 Section 4 loos at the simplest forms of inhomogenous recurrences All of this is well nown, and embodied by the gfun Maple pacage, for example [5, 4, 6 My complementary Maple functions on this theme are available in Date: June 9, 204

2 2 RICHARD J MATHAR 2 Generic Formula The generating function of recurrences [3 s (2) a(n) d i a(n i) + b(n), n n r, i are essentially the generating function of the homogeneous case (b 0) plus the generating function B(x) b(n)xn of the inhomogeneity alone [2: (3) A(x) a(n)x n n a(n)x n + a(n)x n + a(n)x n + a(n)x n nn r s d i a(n i)x n + b(n)x n nn r s d i x i a(n)x n + nn r i i nn r i [a(n) b(n)x n + s i b(n)x n b(n)x n ) n r i d i x (A(x) i a(n)x n + B(x) Separating terms proportional to A(x) and those not depending on A(x) we get (4) ( s i d i x i )A B + [a(n) b(n)x n s i n r i d i x i a(n)x n Dividing through i d ix i generalizes the equation in the abstract to nonzero B 2 -Term Homogeneous Ordered according to increasing distance (stride) s between the indices of the two terms that are coupled with a(n) d s a(n s) we have for example: 2 Stride (5) a(n) d a(n ); a(0) c 0 ; (6) A(x) 22 Stride 2 c 0 d x (7) a(n) d 2 a(n 2); a(0) c 0 ; a() c ; (8) A(x) c 0 + c x d 2 x 2 If d 2 is a positive square, a decomposition in partial fractions might be useful: (9) a(n) 2 2a(n 2); a(0) c 0 ; a() c ; (0) A(x) c 0 + c x ( 2 x)( + 2 x) c 0 2 c x + c c x

3 SOME LINEAR RECURRENCES WITH CONSTANT COEFFICIENTS 3 23 Stride 3 () a(n) d 3 a(n 3); a(0) c 0 ; a() c ; a(2) c 2 ; (2) A(x) c 0 + c x + c 2 x 2 d 3 x 3 If d is a cube, the follow-up decomposition in partial fractions is (3) a(n) 3 3a(n 3); a(0) c 0 ; a() c ; a(2) c 2 ; A(x) (4) c 0 + c x + c 2 x 2 ( 3 x)( + 3 x x2 ) c c 3 c 2 + (c c 2 3 2c 2 3 )x + 3 x x c c 3 + c 2 3 x 24 Stride 4 (5) a(n) d 4 a(n 4); a(0) c 0 ; a() c ; a(2) c 2 ; a(3) c 3 ; (6) A(x) c 0 + c x + c 2 x 2 + c 3 x 3 d 4 x 4 The case of d being a square is in particular (7) a(n) 2 4a(n 4); a(0) c 0 ; a() c ; a(2) c 2 ; a(3) c 3 ; (8) A(x) c 0 + c x + c 2 x 2 + c 3 x 3 ( 4 x 2 )( + 4 x 2 ) 25 General The obvious pattern is (9) a(n) d s a(n s); a(i) c i ; 0 i < s; (20) A(x) This is the most busy case [: 2 4 c c 2 + (c 4 + c 3 )x 4 x c 0 4 c 2 + (c 4 c 3 )x + 4 x 2 s i0 c ix i d s x s 3 2-Term Homogeneous (2) a(n) d a(n ) + d 2 a(n 2); a(0) c 0 ; a() c ; (22) A(x) c 0 + (c d c 0 )x d x d 2 x 2 The case d 0 reduces to (8) 3 First Term Not Coupled (23) a(n) d 2 a(n 2) + d 3 a(n 3); a(0) c 0 ; a() c ; a(2) c 2 ; (24) A(x) c 0 + c x + (c 2 c 0 d 2 )x 2 d 2 x 2 d 3 x 3

4 4 RICHARD J MATHAR 32 Second Term Not Coupled (25) a(n) d a(n ) + d 3 a(n 3); a(0) c 0 ; a() c ; a(2) c 2 ; (26) A(x) c 0 + (c c 0 d )x + (c 2 c d )x 2 d x d 3 x 3 33 First Two Terms Not Coupled (27) a(n) d 3 a(n 3) + d 4 a(n 4); a(0) c 0 ; a() c ; a(2) c 2 ; a(3) c 3 ; (28) A(x) c 0 + c x + c 2 x 2 + (c 3 c 0 d 3 )x 3 d 3 x 3 d 4 x 4 34 First s Terms Not Coupled (22), (24) and (28) are special cases of (29) a(n) d s a(n s) + d s+ a(n s ); a(i) c i ; 0 i s; s ; (30) A(x) s i0 c ix i c 0 d s x s d s x s d s+ x s+ 35 Bisections If the denominator is a polynomial in a higher power of x, the sequence is an overlay of de-facto decoupled subsequences Consider for example the generating function (3) A(x) c 0 + c x + c 2 x 2 + c 3 x 3 d 2 x 2 d 4 x 4 which has no terms x or x 3 in the denominator This defines two subsequences at even and odd indices of the form (32) (33) a(2n) d 2 a(2n 2) + d 4 a(2n 4); a(2n ) d 2 a(2n 3) + d 4 a(2n 5), with initial values a(0) and a(2) for the even terms and a() and a(3) for the odd terms We show how the 6 parameters [four initial values a(03) and two coefficients d can be reorganized as (34) (35) a(2n) β e a(2n ) + β 2e a(2n 2); a(2n ) β o a(2n 2) + β 2o a(2n 3), with 6 parameters [four coefficients β and 2 initial values a(0) and a() that mix the two subsequences The β are obtained as follows Splitting A(x) in the even function (c 0 +c 2 x 2 )/( d 2 x 2 d 4 x 4 ) and the odd function (c x+c 3 x 3 )/( d 2 x 2 d 4 x 4 ) generates for even indices a(2n) [x 2n A(x) c 0 [x 2n d 2 x 2 d 4 x 4 + c 2[x 2n d 2 x 2 d 4 x 4 (36) c 0 [x 2n d 2 x 2 d 4 x 4 + c 2[x 2n 2 d 2 x 2 d 4 x 4 ; (37) a(2n 2) c 0 [x 2n 2 d 2 x 2 d 4 x 4 + c 2[x 2n 4 d 2 x 2 d 4 x 4, x 2

5 SOME LINEAR RECURRENCES WITH CONSTANT COEFFICIENTS 5 and for odd indices a(2n ) [x 2n A(x) c [x 2n x d 2 x 2 d 4 x 4 + c 3[x 2n d 2 x 2 d 4 x 4 (38) c [x 2n 2 d 2 x 2 d 4 x 4 + c 3[x 2n 4 d 2 x 2 d 4 x 4 The previous two equations are a linear 2 2 system of equations for [x 2n 2 d 2x 2 d 4x 4 and [x 2n 4 d 2x 2 d 4x which is solved by 4 (39) [x 2n 2 d 2 x 2 d 4 x 4 a(2n 2) c 2 a(2n ) c 3 / c 0 c 2 c c 3 c 3a(2n 2) c 2 a(2n ) ; c 3 c 0 c 2 c (40) [x 2n 4 d 2 x 2 d 4 x 4 c 0 a(2n 2) c a(2n ) / c 0 c 2 c c 3 c 0a(2n ) c a(2n 2) c 3 c 0 c 2 c We insert the generic recurrence for the auxiliary sequence, 0, d 2, 0, d 4, 0, d d 4, 0, d d 2 d 4,, (4) [x 2n d 2 x 2 d 4 x 4 d 2[x 2n 2 d 2 x 2 d 4 x 4 + d 4[x 2n 4 d 2 x 2 d 4 x 4 in the right hand side of (36) (42) a(2n) (c 0 d 2 + c 2 )[x 2n 2 d 2 x 2 d 4 x 4 + c 0d 4 [x 2n 4 d 2 x 2 d 4 x 4, and then (39) and (40) (43) (c 0 d 2 + c 2 ) c 3a(2n 2) c 2 a(2n ) c 3 c 0 c 2 c + c 0 d 4 c 0 a(2n ) c a(2n 2) c 3 c 0 c 2 c By comparison with the form (34) we conclude (44) (45) β e c2 0d 4 c 2 2 c 0 d 2 c 2 c 3 c 0 c c 2, β 2e c 3c 0 d 2 c c 0 d 4 + c 3 c 2 c 3 c 0 c c 2 The equivalent computation for the odd indices yields x 3 (46) (47) β o β 2o c 2 d 4 c c 3 d 2 c 2 3, c c 0 d 4 c 3 c 2 c 3 c 0 d 2 d 4 (c 3 c 0 c c 2 ) c c 0 d 4 c 3 c 2 c 3 c 0 d 2 4 Inhomogeneous With (4), calculation of A(x) reduces to the calculation of B(x), that is, to looing at the simpler format (48) a(n) b(n)

6 6 RICHARD J MATHAR 4 Simple Powers If b(n) is a linear combination of nth powers with constant coefficients with optional offsets o j, (49) a(n) j0 d j b n oj j, where neither the d j nor the b j nor the o j depend on n, the generating function is the associated geometric series [, 30 (50) A(x) j0 d j b oj j b j x 42 Polynomials The case of the constant term (5) a(n) is the simplest form of (50) with the generating function [, 360 (52) A(x) x -fold differentiation with respect to x computes the generating functions of th order polynomials of n of the format (53) a(n) n(n )(n 2) (n + ) n!/(n )!, (54) A(x)!x ; 0,, 2, ( x) + (See [7, () for the determination of the exponential generating function along the same lines) Decomposition of the general th order polynomial into a sum of polynomials of this special ind by aid of the Stirling Numbers of the Second Kind S [, 244 pairs the polynomial (55) b(n) j0 e j n j with constant coefficients e j with the generating function (56) A(x) j0 e j j 0 S () x j! ( x) + The same methodology of repeated differentiation with respect to x may be applied to the more general (50) and allows construction of generating functions for a(n) j0 0 e,jn b n j, sums of products of simple powers and polynomials 5 Transformation of Series 5 Multisection, Delta-Operator Generating functions of multisections of sequences and the inverse which is a shuffling operation of many sequences into one [6 first and higher order differences are implemented as described by Riordan [5

7 SOME LINEAR RECURRENCES WITH CONSTANT COEFFICIENTS 7 52 Binomial Transform The (inverse) binomial transform relates two sequences a(n) and b(n) via ( ) n ( ) n (57) a(n) b(); b(n) ( ) n a();, 0 0 which induces a relation between the generating functions A(x) n a(n)xn and B(x) n b(n)xn as follows [3, 7, 8, 2, 20: ( ) n ( ) n A(x) b()x n b()x n 0 0 n ( ) s + ( ) s + b()x s+ b()x x s 0 s0 0 s0 b()x ( x) + ( ) x b() ( ) x (58) x x x B x (59) B(x) + x A 0 0 ( x + x These methods can be chained to provide generating functions of other transforms [8, 9, 22 References Milton Abramowitz and Irene A Stegun (eds), Handboo of mathematical functions, 9th ed, Dover Publications, New Yor, 972 MR (29 #494) 2 Roberto Bagnara, Parma university s recurrence relation solver, arxiv:cs/ Mira Bernstein and Neil J A Sloane, Some canonical sequences of integers, Lin Alg Applic (995), 57 72, (E:) [4 MR (96i:05004) 4 Richard A Brualdi, From the editor-in-chief, Lin Alg Applic 320 (2000), no 3, MR Huantian Cao, Autogf: an automated system to calculate coefficients of generating functions, Master s thesis, Massachusetts Institute of Technology, Wenchang Chu, Some binomial convolution formulas, Fib Quart 40 (2002), no, Ayhan Dil and István Mezö, A symmetric algorithm for hyperharmonic and fibonacci numbers, arxiv: [mathnt (2008) 8, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl Math Comput 206 (2008), no 2, MR Asho Kumar Gupta and Asho Kumar Mittal, Integer sequences associated with integer monic polynomial, arxiv:mathgm/0002 (2000) 0 Subhash Ka, The golden mean and the physics of aesthetics, arxiv:physics/0495 (2004) István Mező, Several generating functions for second-order recurrence sequences, J Integer Seq (2009), no 2, 0938 MR (200c:09) 2 Tang Minh and Tan Van To, Using generating functions to solve linear inhomogeneous recurrence equations, Int Conf Simulation, Modelling and Optimization, vol 6, 2006, p G Myerson and A J van der Poorten, Some problems concerning recurrence sequences, Amer Math Monthly 02 (995), no 8, MR (97a:029) 4 Simon Plouffe, 03 generating functions and conjectures, John Riordan, Combinatorial identities, John Wiley, New Yor, 968 MR (38 #53) 6 Bruno Salvy and Paul Zimmerman, Gfun: a maple pacage for the manipulation of generating and holonomic functions in one variable, ACM Trans Math Softw 20 (994), no 2, Susumu Shirai and Ken ichi Sato, Some identities involving Bernoulli and Stirling numbers, J Number Theory 90 (200), no, Michael Z Spivey, Combinatorial sums and finite differences, Discrete Math 307 (2007), no 24, MR (2008j:0503) )

8 8 RICHARD J MATHAR 9 Michael Stoll, Bounds for the length of recurrence relations for convolutions of p-recursive sequences, Eur J Comb 8 (997), no 6, 707 MR (99f:05007) 20 Zhi-Hong Sun, Invariant sequences under binomial transformation, Fib Quart 39 (200), no 4, MR 8553 (2002f:02) 2 Stefan Weinzierl, Expansion around half-integer values, binomial sums, and inverse binomial sums, J Math Phys 45 (2004), no 7, MR (2005f:33042) 22 P Wynn, A note on the generalised Euler transformation, Comp J 4 (97), no 4, MR (47 #9799) URL: Max-Planc Institute of Astronomy, Königstuhl 7, 697 Heidelberg, Germany