GENERALIZED PASCAL TRIANGLES AND TOEPLITZ MATRICES

GENERALIZED PASCAL TRIANGLES AND ¨ TOEPLITZ MATRICES∗

arXiv:0901.2597v1 [math.RA] 19 Jan 2009

A. R. Moghaddamfar and S. M. H. Pooya Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology, P. O. Box 16315 − 1618, Tehran, Iran and Institute for Studies in Theoretical Physics and Mathematics (IPM) E-mail address: [email protected] and [email protected]. January 19, 2009

Abstract The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a T¨ oeplitz matrix and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant of a T¨ oeplitz matrix. This equality allows us to evaluate a few determinants of generalized Pascal matrices associated to certain sequences. In particular, we obtain families of quasi-Pascal matrices whose principal minors generate any arbitrary linear subsequences F (nr + s) or L(nr + s), (n = 1, 2, 3, . . .) of Fibonacci or Lucas sequence.

Key words: Determinant, Matrix factorization, Generalized Pascal triangle, Generalized symmetric (skymmetric) Pascal triangle, T¨ oeplitz matrix, Fibonacci (Lucas, Catalan) sequence, Golden ratio. AMS subject classifications. 15A15, 11C20.

1

Introduction


Let P (∞) be the infinite symmetric matrix with entries Pi,j = i+j for i, j ≥ 0. The matrix P (∞) is hence i the famous Pascal triangle yielding the binomial coefficients. The entries of P (∞) satisfy the recurrence relation Pi,j = Pi−1,j + Pi,j−1 . Indeed, this matrix has the following form:   1 1 1 1 1 ...  1 2 3 4 5 ...     1 3 6 10 15 . . .    (1) P (∞) =  1 4 10 20 35 . . .     1 5 15 35 70 . . .    .. . . .. .. .. .. . . . . . . One can easily verify that (see [2, 10]):

P (∞) = L(∞) · L(∞)t , ∗ This

research was in part supported by a grant from IPM (No. 85200038).

1

(2)

where L is the infinite unipotent lower triangular  1  1   1  L(∞) =  1   1  .. .

matrix 1 2 3 4 .. .

1 3 1 6 4 .. .. . .

 1 .. .

..

.

       

(3)


with entries Li,j = ji . To introduce the result, we first present some notation and definitions. We recall that a matrix T (∞) = (ti,j )i,j≥0 is said to be T¨ oeplitz if ti,j = tk,l whenever i − j = k − l. Let α = (αi )i≥0 and β = (βi )i≥0 be two sequences with α0 = β0 . We shall denote by Tα,β (∞) = (ti,j )i,j≥0 the T¨oeplitz matrix with αi = ti,1 and βj = t1,j . We also denote by Tα,β (n) the submatrix of Tα,β (∞) consisting of the elements in its first n rows and columns. We come now back to the Eq. (2). In fact, one can rewrite it as follows: P (∞) = L(∞) · I · L(∞)t , where matrix I (identity matrix) is a particular case of a T¨oeplitz matrix. In [1], Bacher considers determinants of matrices generalizing the Pascal triangle P (∞). He introduces generalized Pascal triangles as follows. Let α = (αi )i≥0 and β = (βi )i≥0 be two sequences starting with a common first term α0 = β0 = γ. Then, the generalized Pascal triangle associated to α and β, is the infinite matrix Pα,β (∞) = (Pi,j )i,j≥0 with entries Pi,0 = αi , P0,i = βi and Pi,j = Pi−1,j + Pi,j−1 , for i, j ≥ 1. We denote by Pα,β (n) the finite submatrix of Pα,β (∞) with entries Pi,j , 0 ≤ i, j ≤ n − 1. An explicit formula for entry Pi,j of Pα,β (n) is also given by the following formula (see [1]): Pi,j = γ

  X     j i i+j i+j −s  X i+j−t  + (αs − αs−1 ) + (βt − βt−1 ) . j j i s=1 t=1

For arbitrary sequence α = (αi )i≥0 , we define the sequences α ˆ = (ˆ αi )i≥0 and α ˇ = (ˇ αi )i≥0 as follows: α ˆi =

i X

k=0

i+k

(−1)

i   X i and α ˇi = αk . k

  i αk k

(4)

k=0

With these definitions we can now state our main result. Indeed, the purpose of this article is to obtain a factorization of the generalized Pascal triangle Pα,β (n) associated to the arbitrary sequences α and β, as a product of a unipotent lower triangular matrix L(n), a T¨oeplitz matrix Tα, ˆ βˆ (n) and a unipotent upper triangular matrix U (n) (see Theorem 1), that is Pα,β (n) = L(n) · Tα, ˆ βˆ (n) · U (n). Similarly, we show that −1 Tα,β (n) = L(n)−1 · Pα, . ˇ βˇ (n) · U (n)

In fact, we obtain a connection between generalized Pascal triangles and T¨oeplitz matrices. In view of these factorizations, we can easily see that det(Pα,β (n)) = det(Tα, ˆ βˆ (n)). Finally, we present several applications of Theorem 1 to some other determinant evaluations. We conclude the introduction with notation and terminology to be used throughout the article. By ⌊x⌋ we denote the integer part of x, i. e., the greatest integer that is less than or equal to x. We also denote by ⌈x⌉ the smallest integer greater than or equal to x. Given a matrix A, we denote by Ri (A) and Cj (A) the 2

row i and the column j of A, respectively. We use the notation At for the transpose of A. Also, we denote by ,...,jk Aji11,i,j22,...,i l the submatrix of A obtained by erasing rows i1 , i2 , . . . , il and columns j1 , j2 , . . . , jk . In general, an n × n matrix of the following form:     B B A A resp. C Pα,β (n − k) C Tα,β (n − k) where A, B and C are arbitrary matrices of order k × k, k × (n − k) and (n − k) × k, respectively, is called a quasi-Pascal (resp. quasi-T¨ oeplitz) matrix. Throughout this article we assume that: F = (Fi )i≥0 = (0, 1, 1, 2, 3, 5, 8, . . . , Fi = F (i), . . .)

(Fibonacci numbers),

L = (Li )i≥0 = (2, 1, 3, 4, 7, 11, 18, . . . , Li = L(i), . . .)

(Lucas numbers),

∗

F = (Fi )i≥1 = (1, 1, 2, 3, 5, 8, . . . , Fi = F (i), . . .)

C = (Ci )i≥0 = (1, 1, 2, 5, 14, 42, 132, . . ., Ci = C(i), . . .)

(Fibonacci numbers 6= 0),

(Catalan numbers),

I = (Ii )i≥0 = (0!, 1!, 2!, 3!, 4!, 5!, 6!, . . . , Ii = i!, . . .)

I ∗ = (Ii )i≥1 = (1!, 2!, 3!, 4!, 5!, 6!, . . . , Ii = i!, . . .)

The paper is organized as follows: In Section 2, we derive some preparatory results. In Section 3, we prove the main result (Theorem 1) and in Section 4, we present some applications of main theorem.

2

Preliminary Results

As we mentioned in Introduction, if α = (αi )i≥0 is an arbitrary sequence, then we define the sequences α ˆ = (ˆ αi )i≥0 and α ˇ = (ˇ αi )i≥0 as in Eq. (4). For some certain sequences α the associated sequences α ˆ and α ˇ seem also to be of interest since they have appeared elsewhere. In Tables 1 and 2, we have presented some sequences α and the associated sequences α ˆ and α ˇ. Table 1. Some well-known sequences α and associated sequences α. ˇ α (0, 1, −1, 2, −3, 5, −8, . . .) (1, 0, 1, −1, 2, −3, 5, −8, . . .) (2, −1, 3, −4, 7, −11, 18, . . .) (1, 0, 1, 1, 3, 6, 15, . . .) (1, 0, 1, 2, 9, 44, 265, . . .) (1, 1, 3, 11, 53, 309, 2119, . . .)

Reference A039834 in [7] A039834 in [7] A061084 in [7] A005043 in [7] A000166 in [7] A000255 in [7]

α ˇ F F∗ L C I I∗

Table 2. Some well-known sequences α and associated sequences α. ˆ α (0, 1, 3, 8, 21, 55, 144, . . .) (1, 2, 5, 13, 34, 89, 233, . . .) (2, 3, 7, 18, 47, 123, 322, . . .) (1, 2, 5, 15, 51, 188, 731, . . .) (1, 2, 5, 16, 65, 326, 1957, . . .) (1, 3, 11, 49, 261, 1631, . . .)

Reference A001906 in [7] A001519 in [7] A005248 in [7] A007317 in [7] A000522 in [7] A001339 in [7]

α ˆ F F∗ L C I I∗

As an another nice example, consider α = Ri (L(∞)) where L is introduced as in Eq. (3). Then, we have α ˇ = Ri (P (∞)).

ˇˆ = α. ˆˇ = α Lemma 1 Let α be a sequence. Then we have α 3

ˆˇ = (α ˆˇ i )i≥0 . Then, we have Proof. Suppose α = (αi )i≥0 and α ˆˇ i α

Pi

i+k i k=0 (−1) k

=

Pi

k s


αs

Pi Pk i+k i k k=0 s=0 (−1) k s αs

P P i i (−1)i l=0 αl h=l (−1)h hi hl
i−l
Pi Pi (−1)i l=0 αl h=l (−1)h il h−l
Pi
P h i−l (−1)i il=0 αl il h=l (−1) h−l . i+k k=0 (−1) k

= = = = = But, if l < i, then we have


α ˇk
i Pk

Pi

h i−l h=l (−1) h−l

ˆˇ i = (−1)i α




s=0

= 0. Therefore, we obtain

i X l=i

αl

 X   i i i−l = αi , (−1)h l h−l h=l

ˆˇ = α. and hence α The proof of second part is similar to the previous case.  Lemma 2 Let i, j be positive integers. Then we have i−j X

(−1)k

k=0



The proof follows from the easy identity 

i k+j

i k+j

  ( 0 k+j = j 1

if

i 6= j,

if

i = j.

     k+j i i−j = . j j k

The following Lemma is a special case of a general result due to Krattenthaler (see Theorem 1 in [9]). Lemma 3 Let α = (αi )i≥0 and β = (βj )j≥0 be two geometric sequences with αi = ρi and βj = σ j . Then, we have det(Pα,β (n)) = (ρ + σ − ρσ)n−1 .

3

Main Result

Now, we are in the position to state and prove the main result of this article. Theorem 1 Let α = (αi )i≥0 and β = (βi )i≥0 be two sequences starting with a common first term α0 = β0 = γ. Then we have Pα,β (n) = L(n) · Tα, (5) ˆ βˆ (n) · U (n), and −1 Tα,β (n) = L(n)−1 · Pα, , ˇ βˇ (n) · U (n)

where L(n) = (Li,j )0≤i,j