Sections and Lacunary Sums of Linearly Recurrent Sequences

Advances in Applied Mathematics 23, 176᎐197 Ž1999. Article ID aama.1998.0640, available online at http:rrwww.idealibrary.com on

Sections and Lacunary Sums of Linearly Recurrent Sequences Luis Verde-Star* Departamento de Matematicas, Uni¨ ersidad Autonoma Metropolitana, ´ ´ Iztapalapa, Mexico E-mail: [email protected] Received September 1, 1998; accepted September 19, 1998

1. INTRODUCTION Linear recurrent sequences have been extensively studied for a long time because of their importance in combinatorics, difference equations, number theory, algebra, etc. Žsee Refs. w1, 4, 5, 7x.. Many combinatorial objects, like combinatorial sums, are linearly recurrent sequences as functions of one of their parameters Žsee w2, 6x.. An important example is the class of lacunary sums of binomial coefficients. Howard and Witt w3x studied recently some aspects of such combinatorial sums using the method of multisection of series. In this paper we study some aspects of the linearly recurrent sequences ŽLRS. related to sums of binomial coefficients. In particular we study certain linear operators on the vector space of LRS that include sections and sums. A section of a sequence F Ž k . is a sequence GŽ k . s F Ž rk q s ., where r and s are fixed integers. The sum of F Ž k . is the sequence 1 Ž . ⌺F Ž m. s Ý my ks 0 F k , and a lacunary sum of F is the sum of a section of F. In our development we use the basis of the space of LRS that consists of the sequences s a, k Ž m. s Ž mk . a my k , for a in the complex numbers and k in the natural numbers. This gives an immediate connection between LRS and binomial coefficients. We also use the fact that the vector space of LRS is isomorphic to the space of proper rational functions. In fact, they are also isomorphic as Hopf algebras w9, 11x. * Research partially supported by a grant from CONACYT-Mexico. ´ 176 0196-8858r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

177

LINEARLY RECURRENT SEQUENCES

In Section 2 we present the background material, which is an algebraic theory of rational functions that we introduced in previous papers w8᎐12x. In Section 3 we describe the relationship between rational functions and LRS. In the remainder of the paper we study sections and sums of LRS and we find general formulas and recurrence relations for several types of sums of binomial coefficients. As a sample of our results see Eqs. Ž5.4., Ž5.10., Ž5.13., Ž6.8., Ž6.10., and Ž6.12. below. We also obtain the multisection of series using our computational tools.

2. RATIONAL FUNCTIONS We denote by P the complex vector space of all polynomials in one complex variable, and for any nonnegative integer n we denote by Pn the subspace of P of the polynomials whose degree is at most equal to n. Define the basic rational functions r a, k Ž z . s

1

Ž z y a.

1qk

a g ⺓,

,

k g ⺞,

Ž 2.1.

and let R be the complex vector space generated by all the r a, k . It is easy to see that R is the set of all rational functions of the form pŽ z .ruŽ z ., where p and u are polynomials, and u is monic and has degree strictly greater than the degree of p. The elements of R are called proper rational functions. Let Q be the space of all the rational functions. By the division algorithm for polynomials, Q is the direct sum of P and R, and thus the union of the sets  z n : n g ⺞4 and  r a, k : a g ⺓, k g ⺞4 is a basis for Q. We define next an antisymmetric bilinear map from Q = Q to ⺓. Define

² ra, k Ž z . , z n: s ² ra, k Ž z . , r b , m Ž z .: s Ž y1

n nyk a , k

ž / . ž /Ž k

kqm k

a g ⺓,

k, n g ⺞,

1 a y b.

1qkqm

,

Ž 2.2a.

a/b, k , ng⺞,

Ž 2.2b. ² r a, k , r a, m : s 0,

a g ⺓,

k, n g ⺞,

Ž 2.2c.

and ² z , z k : s 0,

k , n g ⺞.

Ž 2.2d.

From Ž2.2b. it is clear that ² r a, k , r b, m : s y² r b, m , r a, k :. Therefore, the above definition and ² f, g : s y² g, f :, for f and g in the basis of Q ,

178

LUIS VERDE-STAR

define the map ² , : on the basis of Q , and extending it by linearity we obtain an antisymmetric bilinear form on Q. We list next some properties of this bilinear form. The proofs are simple computations Žsee w9x and w11x.. Let pru be an element of R and let g be in Q. Then p

,g s

¦ ; u

Ý Residue of a

p

ž / u

g at a,

Ž 2.3.

where the sum runs over the set of distinct roots of uŽ z .. Recall that the residue of a rational function h at a point a is the coefficient of Ž z y a.y1 in the Laurent series expansion of hŽ z . in powers of z y a. Let f and g be rational functions. Then ² f , pg : s ² pf , g : ,

p g P,

Ž 2.4.

² f , Dg : s ² y Df , g : ,

Ž 2.5.

where D denotes differentiation with respect to z; n

² r a, n , fg : s

Ý ² ra, k , f :² ra, nyk , g : ,

Ž 2.6.

ks0

and

² f Ž z . , g Ž z .: s² f Ž z q a. , g Ž z q a.: ,

a g ⺓.

Ž 2.7.

Equation Ž2.6. is Leibniz’s rule and Ž2.7. is the translation invariance property. We define the re¨ ersion map R from Q to Q by Rf Ž z . s

1 z

f

1

ž / z

,

f g Q.

Ž 2.8.

Note that Rz n s zy1yn s r 0, n Ž z . ,

n g ⺞,

Ž 2.9.

and Rra, k Ž z . s

zk

Ž 1 y az .

1qk

,

a g ⺓,

k g ⺞.

Ž 2.10.

It is clear that R 2 s I. We also have ² f, g : s ² Rg, Rf : and ² f, Rg : s ² y Rf, g : for any rational functions f and g. Let pru be a proper rational function and let q and ¨ be polynomials. Then p pq r , q¨ s ,¨ s ,¨ , Ž 2.11. u u u

¦ ;¦ ;¦ ;

LINEARLY RECURRENT SEQUENCES

179

where pq s wu q r and the degree of r is strictly smaller than the degree of u. We say that a function f is defined on the roots of a polynomial uŽ z . if for each root a of u with multiplicity m, the derivatives D k f, for 0 F k m, are defined at a. Using Ž2.3. it is easy to prove the following propositions. Let u and ¨ be polynomials and let f be a function that is defined on the roots of u¨ . Then 1

1

¦ ;¦ ; u¨

, ¨f s

u

,f ,

Ž 2.12.

and, if u and ¨ have no common roots, then 1

1

f

1

f

¦ ;¦ ;¦ ; u¨

,f s

,

u ¨

q

,

¨

u

.

Ž 2.13.

These properties are called Popoviciu’s reduction and decomposition formulas, respectively. Let uŽ z . s Ž z y a.1q k and ¨ Ž z . s Ž z y b .1q m , where a / b. Then, by Ž2.13. and Ž2.6., for any function f defined on the roots of u¨ we have k

² r a, k r b , m , f : s

m

² r a, j , r b , m :² r a, kyj , f : q

Ý js0

Ý ² r b , j , ra, k :² r b , myj , f : . js0

Ž 2.14. Taking f Ž z . s 1rŽ t y z ., where t / a and t / b, we get the multiplication formula k

r a, k r b , m s

m

Ý ² ra, j , r b , m : ra, kyj q Ý ² r b , j , ra, k : r b , myj . Ž 2.15. js0

js1

This is the basic decomposition formula for partial fractions. Now let u j Ž z . s Ž z y a j . m j , for 0 F j F s, where the a j are distinct complex numbers and the m j are positive integers. Let w s ⌸ u j and define q j s wru j . Let f be a function defined on the roots of w. Then, using the decomposition formula Ž2.13. repeatedly, we obtain 1

s

1

f

¦ ; ݦ ; w

,f s

js0

,

u j qj

,

180

LUIS VERDE-STAR

and by Leibniz’s rule we get 1

¦ ; w

m jy1

s

,f s

Ý Ý js0 ks0

1

¦ ; ra j , k ,

² r a , m y1yk , f : . j j

qj

Ž 2.16.

Taking f Ž z . s pŽ z .rŽ t y z ., where p is a polynomial and w Ž t . / 0, we obtain the general partial fraction decomposition formula m jy1

s

pŽ t .

s

wŽ t.

Ý Ý js0 ks0

¦

ra j , k ,

p qj

;Ž

1 t y aj .

m jyk

.

Ž 2.17.

The linear functional that sends f to ²1rw, f : is called the di¨ ided difference of f with respect to the roots of w. Note that Ž2.16. gives an explicit expression for this functional as a linear combination of Taylor functionals at the roots of w. If all the multiplicities m j in Ž2.16. are equal to one we get

¦

1

wŽ z.

s

;

, f Ž z. s

f Ž ak .

.

Ž 2.18.

k G 0.

Ž 2.19.

k i q mi y 1 k ai i , ki

Ž 2.20.

Ý ks0

w⬘ Ž a k .

Let n q 1 s Ý i m i . We define hk s

¦

1

wŽ z.

;

, z nq k ,

It is easy to see w10x that s

hk s

ÝŁ is0

ž

/

where the sum runs over the multiindices Ž k 0 , k 1 , . . . , k s . that satisfy Ý i k i s k. Note that h k is a polynomial in the roots of w. If all the multiplicities m i are equal to one then h k is the complete symmetric polynomial of order k in the variables a0 , a1 , . . . , a n . Using the map R we get hk s

¦

1 z

nqkq1

,

1 zw Ž 1rz .

;¦ s

1 z

kq1

,

1 z

nq1

w Ž 1rz .

;

.

Ž 2.21.

This means that the reciprocal of the polynomial z nq 1 w Ž1rz . is the generating function of the sequence h k . If w Ž z . s z nq 1 q b1 z n q b 2 z ny1 q ⭈⭈⭈ qbnq 1 then z nq 1 w Ž1rz . s 1 q b1 z q b 2 z 2 q ⭈⭈⭈ qbnq1 z nq1. Therefore the product of this polynomial by the series h 0 q h1 z q h 2 z 2 q ⭈⭈⭈ is

181

LINEARLY RECURRENT SEQUENCES

equal to one and thus m

Ý bj h myj s ␦ 0, m ,

m G 0.

Ž 2.22.

js0

For k G 0 we solve for h k , using Cramer’s rule on the finite system that corresponds to m s 0, 1, . . . , k, and we get

k

h k s Ž y1 . det

b1 b2 .. .

1 b1 .. .

1 .. .

bky 1 bk

bky2 bky1

bky3 bky2

..

. ⭈⭈⭈ ⭈⭈⭈

,

k G 1. Ž 2.23.

,

k G 1. Ž 2.24.

1 b1

By the symmetry of the system Ž2.22. we also have

k

bk s Ž y1 . det

h1 h2 .. .

1 h1 .. .

1 .. .

h ky 1 hk

h ky2 h ky1

h ky3 h ky2

..

. ⭈⭈⭈ ⭈⭈⭈

1 h1

Note that bk s 0 if k ) n q 1. If w Ž z . has distinct roots a k , from Ž2.18. we get 1

¦ ; w

, w⬘ f s

n

Ý f Ž ak . . ks0

In particular

¦

1

wŽ z.

;

n

Ý akm s ␴m Ž a0 , a1 , . . . , an . .

, w⬘ Ž z . z m s

Ž 2.25.

ks0

These are the power sum symmetric functions. For any rational function g define the difference quotient g w z, t x s

gŽ z. y gŽ t. zyt

.

Ž 2.26.

If w is a polynomial of degree n q 1 then it is easy to see that ww z, t x is a symmetric polynomial in the variables z and t, of degree n in each variable. If w Ž z . s z nq 1 q b1 z n q b 2 z ny 1 q ⭈⭈⭈ qbnq1 ,

182

LUIS VERDE-STAR

we define the Horner polynomials w k of w by w k Ž z . s z k q b1 z ky 1 q b 2 z ky 2 q ⭈⭈⭈ qbk ,

0 F k F n.

A simple computation yields w w z, t x s

n

Ý wk Ž z . t ny k . ks0

The following biorthogonality relation holds w8x:

¦

1

wŽ z.

;

, wny k Ž z . z j s ␦ k j ,

0 F k F n, 0 F j F n.

Ž 2.27.

PROPOSITION 2.1 ŽHermite’s Interpolation.. Let w Ž z . be a monic polynomial of degree n q 1 and let f be a function defined on the roots of w. Then PŽ t. s

¦

w w z, t x wŽ z.

;

, f Ž z.

Ž 2.28.

is the unique polynomial of degree less than or equal to n that interpolates f at the roots of w in the sense of Hermite.

3. LINEARLY RECURRENT SEQUENCES For each pair Ž a, k . in ⺓ = ⺞ define the sequence s a, k Ž m . s² r a, k Ž z . , z m: s

m myk a , k

ž /

m g ⺞.

Ž 3.1.

Let S be the complex vector space generated by all the sequences s a, k . It is clear that S is isomorphic to R as a complex vector space and the map s a, k ª r a, k is a bijection between bases of S and R. Therefore each sequence F in S corresponds to a unique proper rational function f in R such that F Ž m . s² f Ž z . , z m: ,

m g ⺞.

Ž 3.2.

The forward shift operator E is defined by EF Ž m. s F Ž m q 1. for any sequence F. Let F and f be as in Ž3.2. with f s pru. Since Ez m s z mq1 we have uŽ E . z m s uŽ z . z m and hence uŽ E . F Ž m. s

¦

pŽ z . uŽ z .

;

, u Ž z . z m s² p Ž z . , z m: s 0,

183

LINEARLY RECURRENT SEQUENCES

by the reduction property Ž2.12.. This means that F satisfies a homogeneous linear recurrence relation determined by the denominator u. Suppose u is a monic polynomial of degree n q 1 and write p in the form n

pŽ z . s

Ý c k u nyk Ž z . , ks0

where the u k are the Horner polynomials of u. Then, by Ž2.27. we have FŽ j. s

¦

pŽ z . uŽ z .

n

;

, zj s

ks0

1

¦

Ý ck

uŽ z .

;

, z j u nyk Ž z . s c j ,

0 F j F n.

Ž 3.3. This means that the initial values of the sequence F are determined by the numerator p. From Ž3.2. we get

¦

F Ž m . s Rz m , R

pŽ z . uŽ z .

;¦ s

1 z

mq 1

,R

pŽ z . uŽ z .

;

,

m G 0,

Ž 3.4.

and thus F Ž m. is the value of the Taylor functional at zero of order m applied to the rational function RŽ pru.. In other words, RŽ pru. is the ordinary generating function of the sequence F Ž m. and R

pŽ z . uŽ z .

s

Ý F Ž m. z m .

Ž 3.5.

mG0

For any polynomial ¨ Ž z . of degree s we denote by ¨ 䉫 Ž z . the re¨ ersed polynomial of ¨ , defined by ¨ 䉫 Ž z . s z s ¨ Ž1rz .. If p is a polynomial of degree s and u is a polynomial with degree n q 1, with 0 F s F n, then a simple computation yields R

pŽ z . uŽ z .

s z ny s

p䉫 Ž z . u䉫 Ž z .

.

Ž 3.6.

We present next some examples of linearly recurrent sequences and their corresponding generating functions. From Ž3.1. we see that the generating function of the basic sequence s a, k is Rra, k Ž z . s

zk

Ž 1 y az .

1qk

.

Ž 3.7.

184

LUIS VERDE-STAR

If uŽ z . s Ł njs0 Ž z y a j ., where the a j are distinct, then by Ž2.18.

¦

u⬘ Ž z . uŽ z .

n

;

, zm s

Ý a mj s ␴m ,

m G 0,

Ž 3.8.

js0

and therefore Ž u⬘.䉫 ru䉫 is the generating function of the sequence of power sums ␴m . Let w kq 1Ž z . s z Ž z y 1.Ž z y 2. ⭈⭈⭈ Ž z y k .. Then, for fixed k the sequence Sm , k s

¦

1

;

, zm

w kq 1 Ž z .

Ž 3.9.

is linearly recurrent and its generating function is R

1 w kq 1 Ž z .

s

zk

.

Ž 1 y z . Ž 1 y 2 z . ⭈⭈⭈ Ž 1 y kz .

Ž 3.10.

The Sm, k are the Stirling numbers of the second kind Žsee Comtet w2x.. The relationship between the sequence F Ž m. and the corresponding rational function f s pru of Ž3.2. can also be described in terms of Laurent generating functions. Multiplying both sides of Ž3.2. by tym y1 and summing over m we get

Ý F Ž m . tym y1 s mG0

¦Ž .

f z ,

1 tyz

;

s f Ž t. .

Ž 3.11.

This result may also be obtained from Ž3.5., applying the reversion map R to both sides. If F Ž m. s ² f Ž z ., z m :, where f s pru and uŽ0. / 0, we extend the sequence F to the negative integers defining F Ž ym y 1 . s² f Ž z . , zym y1: s² f Ž z . , Rz m: s y

¦

1 z

mq 1

;

, f Ž z. ,

mG0.

Ž 3.12.

Therefore f Ž t. s y

Ý F Ž ym y 1. t m . mG0

Ž 3.13.

185

LINEARLY RECURRENT SEQUENCES

Combining Ž3.11. and Ž3.13. we obtain Popoviciu’s reciprocity w7, Chap. 4x,

Ý F Ž m . tym y1 s y Ý F Ž ym y 1. t m . mG0

Ž 3.14.

mG0

If uŽ0. / 0, u has degree n q 1, and p has degree s, with s F n, then R

pŽ z . uŽ z .

s z ny s

p䉫 Ž z . u䉫 Ž z .

is an element of R since u䉫 has degree n q 1 and the degree of p 䉫 is at most equal to s. In this case we have that

¦

G Ž m . s F Ž ym y 1 . s y R

pŽ z . uŽ z .

;

, zm ,

m G 0,

Ž 3.15.

is a linearly recurrent sequence that satisfies the equation u䉫 Ž E .GŽ m. s 0, for m G 0, and its initial values GŽ0., GŽ1., . . . , GŽ n. may be determined using the recurrence relation uŽ E . F Ž m. s 0 in the reversed direction. We define next a commutative multiplication * on the space S as follows. If F Ž m. s ² f Ž z ., z m : and GŽ m. s ² g Ž z ., z m :, where f and g are elements of R, then F )G is given by

Ž F )G . Ž m . s² f Ž z . g Ž z . , z m: ,

m G 0.

Ž 3.16.

We also have

Ž F )G . Ž m . s² Rz m , R Ž f Ž z . g Ž z . .: s

¦

1 z

mq 1

;

, zRf Ž z . Rg Ž z . . Ž 3.17.

This means that the generating function of F )G is zRf Ž z . Rg Ž z . and therefore we have Ž F )G .Ž0. s 0 and my1

Ž F )G . Ž m . s

Ý F Ž k . GŽ m y 1 y k . ,

m G 1.

Ž 3.18.

ks0

The multiplication formula Ž2.15. for the basic proper rational functions yields k

s a, k ) sb , n s

n

Ý ² ra, j , r b , n : sa, kyj q Ý ² r b , j , ra, k : sb , nyj . Ž 3.19. js0

js0

186

LUIS VERDE-STAR

4. SHIFTS AND r SECTIONS OF LRS Let f s pru be a proper rational function, where u has degree n q 1, and let F Ž m. s ² f Ž z ., z m : be the corresponding linearly recurrent sequence. Let ¨ Ž t . be a polynomial and define GŽ m. s ¨ Ž E . F Ž m. for m G 0. Then, by Ž2.11., we have GŽ m. s

¦

pŽ z . uŽ z .

;¦

, ¨ Ž z. zm s

rŽ z. uŽ z .

;

, zm ,

where p¨ s qu q r and the degree of r is at most equal to n. The polynomial r Ž t . is the Hermite interpolating polynomial of p¨ at the roots of u and it is given by rŽ t. s

¦

u w z, t x uŽ z .

;

, pŽ z . ¨ Ž z . .

Ž 4.1.

Since u w z, t x s

n

Ý

z k u ny k Ž t . ,

ks0

where the u k are the Horner polynomials of u, substitution in Ž4.1. yields n

rŽ t. s

Ý ks0

¦

1

uŽ z .

n

;

, z k p Ž z . ¨ Ž z . u ny k Ž t . s

Ý G Ž k . u nyk Ž t . . Ž 4.2. ks0

Therefore G is an element of S that satisfies the same linear recurrence that F satisfies and whose initial values are the coefficients in the representation of r Ž t . as a linear combination of the Horner polynomials of u. In particular, for ¨ Ž t . s t s we have G Ž m. s E sF Ž m. s F Ž m q s . ,

m G 0,

Ž 4.3.

and its initial values are GŽ k . s F Ž k q s ., for 0 F k F n. In this case we say that G is the shift of order s of the sequence F. Notice that a sequence of the form ¨ Ž E . F, where ¨ is a polynomial, is a linear combination of shifts of F. Let f s pru be as noted earlier and let F be the corresponding LRS. Let r be a positive integer and let s be an integer such that 0 F s - r. The sequence G defined by G Ž m . s F Ž rm q s . ,

m G 0,

Ž 4.4.

187

LINEARLY RECURRENT SEQUENCES

is called a section of F of order r, or an r section of F. We will show next that any r section of a LRS is also a LRS. Since GŽ m. s F Ž rm q s . s E s F Ž rm., it is clearly enough to show that F Ž rm. is a LRS. Since any LRS is a linear combination of the basic sequences s a, k , by linearity it is enough to prove that any r section of s a, k is in S . We have s a, k Ž rm . s

¦

1

Ž z y a.

1qk

;

, z rm s

rm r myk a , k

ž /

m G 0.

Ž 4.5.

By Newton’s interpolation formula for any polynomial p of degree k we have k

pŽ t . s

t , j

ž/

Ý cj js0

Ž 4.6.

where the coefficients are given by cj s

1 jq1

−ž

1 z jq1

<

0 F j F k.

, pŽ z . ,

/

Ž 4.7.

Using Newton’s interpolation for pŽ t . s Ž rkt . we get rt s k

ž /

k

t , j

ž/

Ý cj js0

Ž 4.8.

where j

cj s

Ý Ž y1. jyi is0

j i

ri , k

0 F j F k.

ž /ž /

Ž 4.9.

Therefore k

s a, k Ž rm . s

Ý js0

cj

m r myk a s j

ž /

k

Ý c j a r jyk js0

m j

ž /Ž

ar .

my j

,

and thus k

s a, k Ž rm . s

Ý c j a r jyk sa , j Ž m . . r

js0

Ž 4.10.

188

LUIS VERDE-STAR

This shows that s a, k Ž rm. is a linear combination of the sequences s a r , j Ž m., for 0 F j F k. We also have s a, k Ž rm . s

¦

c j a r jyk

k

Ý js0

r 1qj

Žzya .

;¦

, zm s

qŽ z.

Ž z y ar .

1qk

;

, z m , Ž 4.11.

where k

Ý c j a r jyk Ž z y a r . ky j ,

qŽ z. s

Ž 4.12.

js0

and the coefficients c j are given by Ž4.9.. Therefore

Ž E y arI .

1q k

s a, k Ž rm . s 0,

m G 0.

Suppose that F is the LRS that corresponds to pru, where u has distinct roots a0 , a1 , . . . , a s with multiplicities m 0 , m1 , . . . , m s , respectively. Then, by linearity, it is clear that any r section of F is a LRS that corresponds to a rational function of the form qr¨ , where ¨ Ž z . is the monic polynomial whose roots are a0r , a1r , . . . , a rs with multiplicities m 0 , m1 , . . . , m s , respectively. We consider next a simple example related to a general Fibonacci type sequence. Let a and b be distinct complex numbers and let F Ž m. s

¦

1

Ž z y a. Ž z y b .

;

, zm s

am y b m ayb

,

m G 0. Ž 4.13.

This is the LRS that satisfies the recurrence

Ž E 2 y Ž a q b . E q abI . F Ž m . s 0,

m G 0,

Ž 4.14.

and has initial values F Ž0. s 0 and F Ž1. s 1. For any positive integer r we have

¦

F Ž rm . s

1

Ž z ya. Ž z yb .

,z

;

rm

s

ar m y b r m a yb

m

s

ar y b r Ž ar . y Ž b r . a yb

ar y b r

m

,

and hence G Ž m . s F Ž rm . s

¦

FŽ r.

Ž z y ar . Ž z y b r .

;

, zm ,

m G 0, Ž 4.15.

189

LINEARLY RECURRENT SEQUENCES

which satisfies the recurrence r

Ž E 2 y Ž a r q b r . E q Ž ab . I . G Ž m . s 0,

m G 0.

Ž 4.16.

The initial values are GŽ0. s 0 and GŽ1. s F Ž r .. Note that Ž4.16. is equivalent to the ‘‘lacunary’’ recurrence relation r

F Ž rm q 2 r . s Ž a r q b r . F Ž rm q r . y Ž ab . F Ž rm . ,

m G 0 Ž 4.17.

Žsee Riordan w6x and Comtet w2x.. If s is a nonnegative integer then it is easy to show that F Ž rm q s . s

¦

qŽ z.

Ž z y ar . Ž z y b r .

;

, zm ,

Ž 4.18.

where qŽ z. s

1 ayb

 as Ž z y b r . y b s Ž z y ar . 4 .

Ž 4.19.

5. PARTIAL SUMS OF THE TERMS OF A LRS We define the sum operator ⌺ on the vector space of all the complexvalued sequences as follows. For any sequence F the sum ⌺F is the sequence determined by ⌺F Ž0. s 0 and my1

⌺F Ž m . s

Ý FŽ j. ,

m G 1.

Ž 5.1.

js0

It is easy to see that Ž E y I . ⌺ s I on the space of all sequences, but ⌺Ž E y I . / I, since Ž ⌺Ž E y I . F .Ž m. s F Ž m. y F Ž0., for m G 0. Let F be a LRS. Then there is a nonzero polynomial u such that uŽ E . F s 0. Therefore uŽ E .Ž E y I . ⌺F s 0 and thus ⌺F is also a LRS. Let us compute the sum ⌺ s a, k of a basic element of S with a / 1. Since s a, k Ž j . s ² r a, k Ž z ., z j : for j G 0, for m G 1 we have

¦

⌺ s a, k Ž m . s r a, k Ž z . ,

my1

Ý js0

;¦

z j s r a, k Ž z . ,

1 y zm 1yz

;

.

Ž 5.2.

190

LUIS VERDE-STAR

Using Leibniz’s rule Ž2.6. we obtain k

⌺ s a, k Ž m . s

Ý js0

¦

r a, kyj Ž z . ,

1 1yz

;²

r a, j Ž z . , 1 y z m:

k

s

Ý ra, kyj Ž 1.  ␦ 0 , j y sa, j Ž m . 4 js0 k

s r a, k Ž 1 . y

Ý ra, kyj Ž 1. sa, j Ž m . . js0

Therefore we have ⌺ s a, k Ž m . s

1

Ž 1 y a.

1qk

½

k

1y

Ý Ž 1 y a. j sa, j Ž m . js0

5

a / 1,

,

m G 1.

Ž 5.3. This equation gives ⌺ s a, k as a linear combination of basic elements of S and it is equivalent to my1

Ý js0

1 j jyk a s 1y 1q k k Ž 1 y a.

½

ž/

k

Ý Ž 1 y a. j js0

m myj a , j

ž /

a/1,

5

mG1.

Ž 5.4.

Note that the number of terms in the sum of the right hand side is k q 1, and it is independent of m. For the case a s 1 we get from Ž5.2. ⌺ s1, k Ž m . s s s

¦ ¦ ž

1

Ž z y 1.

; ;

1qk

2q k

, zm y 1

1

Ž z y 1.

zm y 1

,

zy1

m . kq1

/

This is the well-known identity my1

Ý js0

j m s , k q 1 k

ž/ ž

/

m G 1,

k G 0,

Ž 5.5.

191

LINEARLY RECURRENT SEQUENCES

which is a trivial consequence of the basic recurrence for the binomial coefficients. From the above results, or using Popoviciu’s reduction formula Ž2.12., it is easy to verify that ⌺ s a, k Ž m . s s

¦ ¦

1

Ž z y a.

1qk

Ž z y a.

1q k

Ž z y 1.

1

Ž z y 1.

;

, zm y 1

;

a g ⺓,

, zm ,

k g ⺞. Ž 5.6.

Let F Ž m. s ² f Ž z ., z m : be the LRS that corresponds to the proper rational function f s pru. Since pru is a finite linear combination of basic functions r a, k , by Ž5.6. and linearity we obtain ⌺F Ž m . s

¦

pŽ z . u Ž z . Ž z y 1.

;

m G 0.

, zm ,

Ž 5.7.

Note that, if uŽ1. / 0 and pŽ1. s 0 then ⌺F satisfies the same linear recurrence that F satisfies. Note also that ÝF s F )1, where 1 denotes the constant sequence with value equal to 1. We consider next the computation of lacunary partial sums of terms of a LRS. Given a RLS F and integers r and s such that 0 F s - r, we want a 1 Ž . closed form for Ý my js0 F rj q s . We can use the results of the previous section to write F Ž rj q s . as a linear combination of sequences s a, k Ž j . and then use Ž5.4. and Ž5.5. to find the sum of each s a, k Ž j .. An alternative procedure is to decompose F Ž m. as a linear combination of sequences 1 Ž . s a, k Ž m. and then compute Ý my js0 s a, k rj q s following the method that we used previously to obtain Ž5.3.. In this way we get my1

Ý

my1

¦

s a, k Ž rj q s . s r a, k Ž z . ,

js0

Ý

¦

s r a, k Ž z . , z s k

s

Ý js0

¦

;

z r jqs

js0

1 y z rm 1 y zr

r a, kyj Ž z . ,

;

zs 1 y zr

;²

k

s

Ý c j  ␦ 0, j y sa, j Ž rm . 4 , js0

r a, j Ž z . , 1 y z r m:

192

LUIS VERDE-STAR

where

¦

c j s r a, kyj Ž z . ,

zs 1yz

;¦ s

r

1 z y1 r

;

, z s r a, kyj Ž z . .

Ž 5.8.

Let ⑀ s e i2 ␲ r r. Then, if a / e l for 0 F l F r y 1, then z s r a, kyj Ž z . is well-defined on the roots of z r y 1 and using Ž2.18. we get cj s

1 r

⑀ lŽ sq1.

ry1

Ý

Ž ⑀ l y a.

ls0

1qkyj

.

Ž 5.9.

Therefore we have proved that if a is not an r th root of one then my1

Ý js0

ž

rj q s r jqsyk a s c0 y k

/

k

Ý cj js0

rm r myj a , j

m G 1, Ž 5.10.

ž /

where the coefficients c j are given in Ž5.9.. Note that the right hand side is an r section of a LRS. Suppose now that a r s 1 with r G 2, then a s ⑀ l for some integer l such that 0 F l - r. Writing q Ž z . s Ž1 y z r .rŽ ⑀ l y z . we have my1

Ý

s a, k Ž rj q s . s

js0

s

¦ ¦

1

Ž z y ⑀ l.

,

2q k

,

y1

Ž z y ⑀ l.

kq1

s

1qk

Ý js0

¦

Ý cj js0

Ž1 y z r . qŽ z.

l 2q kyj

Žzy⑀ .

½

␦ j, 0 y

; ; ;¦

z sŽ1 y z rm .

1

kq1

s

z sŽ1 y z rm .

,

yz s qŽ z.

1 l 1qj

Žzy⑀ .

;

, 1 y z rm

rm lŽ r myj. ⑀ , j

5

ž /

where

cj s

¦

1 qŽ z.

,

zs

Ž z y ⑀ l.

2qkyj

;

s

⑀ is

ry1

Ý is0 i/l

Ž⑀ i y ⑀ l.

2qkyj

q⬘ Ž ⑀ i .

. Ž 5.11.

193

LINEARLY RECURRENT SEQUENCES

For example, for a s 1 we have cj s

⑀ iŽ sq1.

ry1

1

Ý

r

Ž ⑀ i y 1.

is1

1qkyj

0 F j F k q 1,

,

Ž 5.12.

and my1

Ý js0

ž

rj q s s k

/

kq1

Ý cj js0

½

␦ j, 0 y

rm j

ž /5

.

Ž 5.13.

6. LACUNARY SUMS OF LRS Let a be a complex number and let r and s be integers such that 0 F s - r. Define n

Sn s

n G 0.

Ý sa, r kqs ,

Ž 6.1.

ks0

It is clear that S n is a LRS and Sn Ž m . s

¦

n

1

Ý

ks0

Ž z y a.

1qsqk r

;

, zm .

Using the formula for the sum of a geometric progression we get Sn Ž m . s

¦

pŽ z .

Ž z y a.

1qsqr n

;

, zm ,

Ž 6.2.

where

Ž z y a. r nq1 y 1 p Ž z . s Ý Ž z y a. s . r Ž z y a. y 1 ks0 Ž

n

.

rk

Ž 6.3.

Note that pŽ a. / 0 and thus pŽ z .rŽ z y a.1q sqr n is an irreducible proper rational function. This means that as a function of m it is not possible to simplify SnŽ m.. From Ž6.2. it is clear that S n satisfies the recurrence relation

Ž E y aI .

1q sqr n

Sn Ž m . s 0,

m G 0.

Ž 6.4.

194

LUIS VERDE-STAR

Using translation invariance in Ž6.2. we obtain Sn Ž m . s

¦

z r nqr y 1 z

1qsqr n

Ž z y 1. r

, Ž z q a.

m

;

,

which may be written as Sn Ž m . s

¦

1

Ž z y 1. r

, z rysy1 Ž z q a .

m

;¦ y

1 z

1qsqr n

Ž z r y 1.

, Ž z q a.

m

;

.

Ž 6.5. Let us consider now that m is a fixed positive integer. Note that the first term in the right hand side of Ž6.5. does not depend on n, and the other term becomes zero for sufficiently large n. More precisely, for n that satisfies Ž n q 1. r ) m y s. Therefore we have Sn Ž m . s

¦

1

Ž z y 1. r

, z rysy1 Ž z q a .

m

;

,

Ž n q 1 . r ) m y s. Ž 6.6.

Let us denote by SŽ m. the common value of SnŽ m. for Ž n q 1. r ) m y s. Note that S Ž m. s

Ý jG0

ž

m a my r jys . rj q s

/

Ž 6.7.

A simple computation yields S Ž m. s

1

ry1

Ý ⑀yjs Ž ⑀ j q a.

r

m

,

Ž 6.8.

js0

where ⑀ s e i2 ␲ r r. From Ž6.6. it is clear that SŽ m. satisfies the recurrence relation

Ž Ž E y aI .

r

y I . S Ž m . s 0,

m G 0.

Ž 6.9.

If a is real and s s 0 then Ž6.8. can be written as S Ž m. s

1 r

½

m

l

Ž a q 1 . q 2 Ý Re  Ž a q ⑀ j . q P Ž r . Ž a y 1 . js1

m

4

5

,

195

LINEARLY RECURRENT SEQUENCES

where P Ž r . is equal to 1 if r is even and to zero if r is odd, and l is Ž r y 2.r2 if r is even and is Ž r y 1.r2 if r is odd. For a s 1 we get S Ž m. s

Ý jG0

l 2m m m s 1 q 2 Ý cos Ž jm␲rr . Ž cos Ž j␲rr . . , Ž 6.10. rj r js1

½

ž /

5

where l is as above. Let us consider now a more general lacunary sum of binomial coefficients. Let m, s, r, and q be integers such that m ) s G 0 and r ) q G 0. Define LŽ m. s

ž

Ý kG0

m q qk . s q rk

/

Ž 6.11.

Then we have LŽ m. s

Ý kG0

s

¦

¦Ý

kG0

s

¦

1

Ž z y 1.

1q sqr k

z qk

Ž z y 1.

1q sqr k

Ž z y 1. Ž z y 1.

1qs

;

, z mq qk

;

, zm

r

 Ž z y 1. r y z q 4

;

, zm .

This shows that LŽ m. is a LRS. If in addition to the hypothesis given above we have also r ) s then rysy1

LŽ m. s

¦

Ž z y 1. , zm , r Ž z y 1. y z q

;

Ž 6.12.

and it satisfies a linear recurrence of order r. Finding the roots of the denominator Ž z y 1. r y z q we can write LŽ m. as a linear combination of basic LRS. Let us consider a simple example. Take r s 2, q s 1, and s s 1. Then the denominator becomes Ž z y 1. 2 y z and its roots are a s Ž3 q '5 .r2 and b s Ž3 y '5 .r2. Therefore LŽ m. s

am y b m ayb

,

m G 0,

and satisfies the recurrence L Ž m q 2. s 3 L Ž m q 1. y L Ž m . , with initial conditions LŽ0. s 0 and LŽ1. s 1.

196

LUIS VERDE-STAR

Finally, we use our methods to obtain the multisection of a formal power series. See Comtet w2x, Riordan w6x, and Howard and Witt w3x. Let f Ž t . s Ý jG 0 bj t j. Then bj t j s

¦

1 z

;

j G 0.

, f Ž tz . ,

1q j

Let r and s be integers such that r ) s G 0. Define the Ž r, s . section of f by fr , s Ž t . s

Ý br jqs t r jqs . jG0

Then we have fr , s Ž t . s

s

s

¦ ¦ ¦

s

1 r

1

Ý jG0

z

1 z

1q s

1qsqr j

; ;

, f Ž tz .

Ý zyj r , f Ž tz . jG0

z rysy1 zr y 1

;

, f Ž tz .

ry1

Ý ⑀yj s f Ž ⑀ j t . ,

js0

where ⑀ s e i2 ␲ r r. The main property of the sections of f is the following: ry1

Ý fr , s Ž t . s ss0

¦

z ry1 z y1 r

ry1

;¦

Ý zys , f Ž tz .

ss0

s

1 zy1

;

, f Ž tz . s f Ž t . .

REFERENCES 1. L. Cerlienco, M. Mignotte, and F. Piras, Suites recurrentes lineaires, Enseign. Math. 33 ´ ´ Ž1987., 67᎐108. 2. L. Comtet, ‘‘Advanced Combinatorics,’’ Reidel, Dordrecht, 1974. 3. F. T. Howard and R. Witt, Lacunary sums of binomial coefficients, in ‘‘Applications of Fibonacci numbers,’’ ŽG. E. Bergum et al., Eds.., Vol. 7, pp. 185᎐195, Kluwer Academic, Dordrecht, 1998.

LINEARLY RECURRENT SEQUENCES

197

4. R. E. Mickens, ‘‘Difference Equations,’’ Van Nostrand᎐Reinhold, New York, 1987. 5. B. Peterson and E. J. Taft, The Hopf algebra of linearly recursive sequences, Aequationes Math. 20 Ž1980., 1᎐17. 6. J. Riordan, ‘‘Combinatorial Identities,’’ Wiley, New York, 1968. 7. R. P. Stanley, ‘‘Enumerative Combinatorics,’’ Vol. I, BrooksrCole, Monterey, CA, 1986. 8. L. Verde-Star, Biorthogonal polynomial bases and Vandermonde-like matrices, Stud. Appl. Math. 95 Ž1995., 269᎐295. 9. L. Verde-Star, A Hopf algebra structure on rational functions, Ad¨ . Math. 116 Ž1995., 377᎐388. 10. L. Verde-Star, Divided differences and linearly recurrent sequences, Stud. Appl. Math. 95 Ž1995., 433᎐456. 11. L. Verde-Star, An algebraic approach to convolutions and transform methods, Ad¨ . in Appl. Math. 19 Ž1997., 117᎐143. 12. L. Verde-Star, Taylor functionals and the solution of linear difference equations, in ‘‘Applications of Fibonacci Numbers,’’ ŽG. E. Bergum et al., Eds.., Vol. 7, pp. 449᎐462, Kluwer Academic, Dordrecht, 1998.