aa r X i v : . [ m a t h . N T ] D ec Fractal generalized Pascal matrices
E. Burlachenko
Abstract
Set of generalized Pascal matrices whose elements are generalized binomial coef-ficients is considered as an integral object. The special system of generalized Pascalmatrices, based on which we are building fractal generalized Pascal matrices, is in-troduced. Pascal matrix (Pascal triangle) is the Hadamard product of the fractalgeneralized Pascal matrices whose elements equal to p k , where p is a fixed primenumber, k = 0 , , , . . . .The concept of zero generalized Pascal matrices, an example of which is the Pascaltriangle modulo 2, arise in connection with the system of matrices introduced. Consider the following generalization of the binomial coefficients [1]. For the coefficientsof the formal power series b ( x ) , b = 0 ; b n = 0 , n > , denote b ! = 1 , b n ! = n Y m =1 b m , (cid:18) nm (cid:19) b = b n ! b m ! b n − m ! ; (cid:18) nm (cid:19) b = 0 , m > n. Then (cid:18) nm (cid:19) b = (cid:18) n − m − (cid:19) b + b n − b m b n − m (cid:18) n − m (cid:19) b .n -th coefficient of the series a ( x ) , ( n, m ) -th element of the matrix A , n -th row and n -thcolumn of the matrix A will be denoted respectively by [ x n ] a ( x ) , ( A ) n,m , [ n, → ] A, [ ↑ , n ] A. We associate rows and columns of matrices with the generating functions of their elements.For the elements of the lower triangular matrices will be appreciated that ( A ) n,m = 0 , if n < m . Consider matrix P c ( x ) = c c c . . . c c c c c c . . . c c c c c c c c c . . . c c c c c c c c c c c c . . . c c c c c c c c c c c c c c c . . . ... ... ... ... ... . . . . (cid:0) P c ( x ) (cid:1) n,m = c m c n − m c n , c n ∈ R , c n = 0 . Denote [ ↑ , P c ( x ) = b ( x ) . If c = 1 , then c n = c n b n ! , (cid:0) P c ( x ) (cid:1) n,m = (cid:18) nm (cid:19) b . c = 1 . Since P c ( x ) = P c ( ϕx ) , we take for uniqueness that c = 1 . Matrix P c ( x ) will becalled generalized Pascal matrix. If c ( x ) = e x , it turns into Pascal matrix P = . . . . . . . . . . . . . . . ... ... ... ... ... . . . . In Sec. 2. we consider the set of generalized Pascal matrices as a group underHadamard multiplication and introduce a special system of matrices, which implies theconcept of zero generalized Pascal matrices; as an example, we consider the matrix of the q -binomial coefficients for q = − . In Sec. 3. we construct a fractal generalized Pascalmatrices whose Hadamard product is the Pascal matrix. In Sec. 4. we consider the fractalzero generalized Pascal matrices that were previously considered in [7] – [11] (generalizedSierpinski matrices and others). Elements of the matrix P c ( x ) , – denote them (cid:0) P c ( x ) (cid:1) n,m = c m c n − m /c n = ( n, m ) for gener-ality which will be discussed later, – satisfy the identities ( n,
0) = 1 , ( n, m ) = ( n, n − m ) , (1)( n + q, q ) ( n + p, m + p ) ( m + p, p ) = ( n + p, p ) ( n + q, m + q ) ( m + q, q ) , (2) q , p = 0 , , , . . . It means that each matrix P c ( x ) can be associated with the algebra offormal power series whose elements are multiplied by the rule a ( x ) ◦ b ( x ) = g ( x ) , g n = n X m =0 ( n, m ) a m b m − n , that is, if ( A ) n,m = a n − m ( n, m ) , ( B ) n,m = b n − m ( n, m ) , ( G ) n,m = g n − m ( n, m ) , then AB = BA = G : g n = ( n + p, p ) − n X m =0 ( n + p, m + p ) ( m + p, p ) a n − m b m == ( n + q, q ) − n X m =0 ( n + q, m + q ) ( m + q, q ) a n − m b m . The set of generalized Pascal matrix is a group under Hadamard multiplication (wedenote this operation × ): P c ( x ) × P g ( x ) = P c ( x ) × g ( x ) , c ( x ) × g ( x ) = ∞ X n =0 c n g n x n . Introduce the special system of matrices ϕ,q P = q P ( ϕ ) = P c ( ϕ,q,x ) , c ( ϕ, q, x ) = q − X n =0 x n ! (cid:18) − x q ϕ (cid:19) − , q > . c qn + i = 1 ϕ n , ≤ i < q ; c qn − i = 1 ϕ n − , < i ≤ q,c qm + j c q ( n − m )+ i − j c qn + i = ϕ n ϕ m ϕ n − m = 1 , i ≥ j ; = ϕ n ϕ m ϕ n − m − = ϕ, i < j, or ( ϕ,q P ) n,m = 1 , n (mod q ) ≥ m (mod q ) ; = ϕ, n (mod q ) < m (mod q ) . For example, ϕ, P , ϕ, P : . . . . . . ϕ . . . . . . ϕ ϕ . . . . . . ϕ ϕ ϕ . . . . . . ϕ ϕ ϕ ϕ . . . ... ... ... ... ... ... ... ... ... . . . , . . . . . . . . . ϕ ϕ . . . ϕ . . . . . . ϕ ϕ ϕ ϕ . . . ϕ ϕ . . . . . . ... ... ... ... ... ... ... ... ... . . . . Elements of the matrix ϕ,q P × P c ( x ) satisfy the identities (1), (2) for any values ϕ , soit makes sense to consider also the case ϕ = 0 since it corresponds to a certain algebra offormal power series (which, obviously, contains zero divisors; for example, in the algebraassociated with the matrix , P × P c ( x ) the product of the series of the form xa ( x ) iszero). It is clear that in this case the series c ( ϕ, q, x ) is not defined. Matrix ,q P × P c ( x ) and Hadamard product of such matrices will be called zero generalized Pascal matrix. Remark.
Zero generalized Pascal matrix appears when considering the set of gener-alized Pascal matrices P g ( q,x ) : (cid:0) P g ( q,x ) (cid:1) n, = [ x n ] x (1 − x ) (1 − qx ) = n − X m =0 q m , q ∈ R . Here g (0 , x ) = (1 − x ) − , g (1 , x ) = e x . In other cases (the q -umbral calculus [2]), except q = − , g ( q, x ) = ∞ X n =0 ( q − n ( q n − x n , ( q n − n Y m =1 ( q m − , (cid:0) q − (cid:1) ! = 1 . Matrices P g ( q,x ) , P − g ( q,x ) also can be defined as follows: [ ↑ , n ] P g ( q,x ) = x n n Y m =0 (1 − q m x ) − , [ n, → ] P − g ( q,x ) = n − Y m =0 ( x − q m ) . When q = − we get the matrices P g ( − ,x ) , P − g ( − ,x ) : . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... . . . , . . . − . . . − . . . − − . . . − . . . − − − . . . − − . . . ... ... ... ... ... ... ... . . . , g ( − , x ) is not defined. Since (cid:0) P g ( − ,x ) (cid:1) n + i, m + j = (cid:2) x n + i (cid:3) (1 + x ) − j x m + j (1 − x ) m +1 = (cid:18) nm (cid:19) , i ≥ j ; = 0 , i < j ; i, j = 0 , , then P g ( − ,x ) = , P × P c ( x ) , c ( x ) = (1 + x ) e x : c n + i = 1 n ! , ≤ i < c n − i = 1( n − , < i ≤ , (cid:0) P c ( x ) (cid:1) n + i, m + j = (cid:18) nm (cid:19) , i ≥ j ; = n (cid:18) n − m (cid:19) , i < j. A generalization of this matrix is the matrix ,q P × P c ( x ) , c ( x ) = q − X n =0 x n ! e x q , (cid:0) P c ( x ) (cid:1) qn + i,qm + j = (cid:18) nm (cid:19) , i ≥ j ; = n (cid:18) n − m (cid:19) , i < j ; 0 ≤ i, j < q. Each nonzero generalized Pascal matrix is the Hadamard product of the matrices ϕ,q P .Since the first column of the matrix P c ( x ) , – denote it b ( x ) , – is the Hadamard productof the first columns of the matrices ϕ,q P , – denote them ϕ,q b ( x ) : [ x n ] ϕ,q b ( x ) = 1 , n (mod q ) = 0; = ϕ, n (mod q ) = 0 , so that P c ( x ) = P ( b ) × P ( b ) × P ( b /b ) × P ( b ) × P ( b /b b ) × P ( b ) ×× P ( b /b ) × P ( b /b ) × P ( b /b b ) × P ( b ) × P ( b b /b b ) × ... and so on. Let e q is a basis vector of an infinite-dimensional vector space. Mapping ofthe set of generalized Pascal matrices in an infinite-dimensional vector space such that ϕ,q P → e q log | ϕ | is a group homomorphism whose kernel consists of all involutions in thegroup of generalized Pascal matrices, i.e. from matrices whose non-zero elements equalto ± . Thus, the set of generalized Pascal matrices whose elements are non-negativenumbers is an infinite-dimensional vector space. Zero generalized Pascal matrices can beviewed as points at infinity of space. Matrices, which will be discussed (precisely, isomorphic to them), are considered in [3]( p -index Pascal triangle), [4], [5, p. 80-88]. These matrices are introduced explicitly in[6] in connection with the generalization of the theorems on the divisibility of binomialcoefficients. We consider them from point of view based on the system of matrices ϕ,q P .We start with the matrix [2] P = , P × , P × , P × ... × , k P × ... P = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . The first column of the matrix [2] P , – denote it b ( x ) , – is the Hadamard product ofthe series , k b ( x ) , k = 1 , , . . . , [ x n ] , k b ( x ) = 1 , n (cid:0) mod2 k (cid:1) = 0; = 2 , n (cid:0) mod2 k (cid:1) = 0 . It is the generating function of the distribution of divisors k in the series of naturalnumbers: b ( x ) = x + 2 x + x + 4 x + x + 2 x + x + 8 x + x + 2 x + x + 4 x ++ x + 2 x + x + 16 x + x + 2 x + x + 4 x + x + 2 x + x + ... Theorem. b k n + i = b i , < i < k . Proof.
It’s obvious that b n = 2 b n , b n +1 = 1 . Then b ( x ) = x − x + 2 b (cid:0) x (cid:1) = k − X n =0 n x n − x n +1 + 2 k b (cid:16) x k (cid:17) = ∞ X n =0 n x n − x n +1 , or b ( x ) = ∞ X n =0 ( n ) a ( x ) , ( n ) a ( x ) = ∞ X m =0 n x n (2 m +1) , ( n ) a n (2 m +1) = ( n ) a k p +2 n (2 m +1) = 2 n , k > n, and ( n ) a m = 0 in other cases. Hence, ( n ) a k p + i = ( n ) a i , < i < k .Note identities b n +1 ! = b n ! , b n − ! = b n − ! ,b n ! = b n b n − ! = 2 b n b n − ! = 2 b n b n − b n − − ! = 2 b n b ( n − b n − ! = ... = 2 n b n ! ,b k n ! = 2 k − n b k − n ! = 2 k − n k − n b k − n ! = ... = 2( k − ) n b n ! . The series ( x ) = P ∞ n =0 x n /b n ! also is the fractal: c ( x ) = 1 + x + x x x + x + x + x + x + x + x + x ++ x + x + x + x + x + x + x + x + x + x + ..., n = c n +1 = 12 n b n ! = c n n , c ( x ) = (1 + x ) c (cid:18) x (cid:19) = ∞ Y n =0 (cid:18) x n n − (cid:19) . Denote (cid:0) [2] P (cid:1) n,m = b n ! b m ! b n − m ! = (cid:18) nm (cid:19) . Since b k n + i ! = b k n ! b i ! = 2( k − ) n b n ! b i ! , ≤ i < k , then (cid:18) k n + i k m + j (cid:19) = (cid:18) nm (cid:19) (cid:18) ij (cid:19) , i ≥ j. For example, matrices [2] P × , P , [2] P × , P have the form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... . . . . If i < j , we transform a numerator and denominator of the generalized binomialcoefficient (cid:0) k n + i k m + j (cid:1) as follows: b k n + i ! = b k n ! b i ! = b k n b k n − ! b i ! = b k b n b k ( n − k − ! b i ! == b k b n b k ( n − ! b k − ! b i ! = b n b k ( n − ! b k + i ! , where b k n = 2 k b n = b k b n . For denominator: b k m + j ! b k ( n − m )+ i − j ! = b k m ! b j ! b k ( n − m − k + i − j ! = b k m ! b j ! b k ( n − m − ! b k + i − j ! . Thus, (cid:18) k n + i k m + j (cid:19) = b n (cid:18) n − m (cid:19) (cid:18) k + ij (cid:19) = b m +1 (cid:18) nm + 1 (cid:19) (cid:18) k + ij (cid:19) , ≤ i < k , i < j. For example, [2] P − (cid:0) [2] P × , P (cid:1) = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... . . . . (cid:18) n + 12 m + 1 (cid:19) = (cid:18) n + 12 m (cid:19) = (cid:18) n m (cid:19) = (cid:18) nm (cid:19) , (cid:18) n m + 1 (cid:19) = 2 b n (cid:18) n − m (cid:19) = 2 b m +1 (cid:18) nm + 1 (cid:19) means that rows and columns of the matrix [2] P , denote them u n ( x ) and g n ( x ) , form therecurrent sequences: u n ( x ) = u n (cid:0) x (cid:1) + 2 b n xu n − (cid:0) x (cid:1) , u n +1 ( x ) = (1 + x ) u n (cid:0) x (cid:1) ; g n ( x ) = (1 + x ) g n (cid:0) x (cid:1) , g n +1 ( x ) = xg n (cid:0) x (cid:1) + 2 b n +1 g n +1 (cid:0) x (cid:1) . Turning to generalize, consider the matrix [3] P = , P × , P × , P × ... × , k P × ... [3] P = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . We will use the same notation as in the previous example. The first column of the matrix [3] P is the generating function of the distribution of divisors k in the series of naturalnumbers: b ( x ) = x + x + 3 x + x + x + 3 x + x + x + 9 x + x + x + 3 x ++ x + x + 3 x + x + x + 9 x + x + x + 3 x + x + x + ...,b k n = 3 k b n , b n +1 = b n +2 = 1 , b k n + i = b i , < i < k ,b n + i ! = b n ! , ≤ i < b n − i ! = b n − ! , < i ≤ ,b n ! = 3 n b n ! , b k n ! = 3 (cid:18) k − (cid:19) n b n ! ,b ( x ) = (1 + x ) x − x + 3 b (cid:0) x (cid:1) = ∞ X n =0 (cid:0) x n (cid:1) n x n − x n +1 ,c n = c n +1 = c n +2 = c n n , ( x ) = (cid:0) x + x (cid:1) c (cid:18) x (cid:19) = ∞ Y n =0 (cid:18) x n (3 n − / + x n ) n − (cid:19) . If b n ! b m ! b n − m ! = (cid:18) nm (cid:19) , then (cid:18) k n + i k m + j (cid:19) = (cid:18) nm (cid:19) (cid:18) ij (cid:19) , ≤ i < k , i ≥ j ; (cid:18) k n + i k m + j (cid:19) = b n (cid:18) n − m (cid:19) (cid:18) k + ij (cid:19) = b m +1 (cid:18) nm + 1 (cid:19) (cid:18) k + ij (cid:19) , ≤ i < k , i < j. From (cid:18) n + i m + j (cid:19) = (cid:18) nm (cid:19) , i ≥ j ; = 3 b n (cid:18) n − m (cid:19) = 3 b m +1 (cid:18) nm + 1 (cid:19) , i < j, we see that if [ n, → ] [3] P = u n ( x ) , [ ↑ , n ] [3] P = g n ( x ) , then u n ( x ) = u n (cid:0) x (cid:1) + 3 b n (1 + x ) xu n − (cid:0) x (cid:1) ,u n +1 ( x ) = (1 + x ) u n (cid:0) x (cid:1) + 3 b n x u n − (cid:0) x (cid:1) ,u n +2 ( x ) = (cid:0) x + x (cid:1) u n (cid:0) x (cid:1) ; g n ( x ) = (cid:0) x + x (cid:1) g n (cid:0) x (cid:1) ,g n +1 ( x ) = (1 + x ) xg n (cid:0) x (cid:1) + 3 b n +1 g n +1 (cid:0) x (cid:1) ,g n +2 ( x ) = x g n (cid:0) x (cid:1) + 3 b n +1 (1 + x ) g n +1 (cid:0) x (cid:1) . Introduce the notation w m ( x ) = m X n =0 x n , w − ( x ) = 0 . For the matrix [ q ] P = q,q P × q,q P × q,q P × ... × q,q k P × ...,q = 2 , , , . . . , we have (in the same notation as in the previous examples): b q k n = q k b n , b qn + i = 1 , < i < q, b q k n + i = b i , < i < q k .b q k n ! = q (cid:18) qk − q − (cid:19) n b n ! ,b ( x ) = w q − ( x ) x − x q + qb ( x q ) = ∞ X n =0 w q − (cid:0) x q n (cid:1) q n x q n − x q n +1 ; c qn + i = c n q n , ≤ i < q,c ( x ) = w q − ( x ) c (cid:18) x q q (cid:19) = ∞ Y n =0 w q − (cid:16) x q n /q qn − q − (cid:17) ; (cid:18) q k n + iq k m + j (cid:19) q = (cid:18) nm (cid:19) q (cid:18) ij (cid:19) q , ≤ i < q k , i ≥ j, q k n + iq k m + j (cid:19) q = b n (cid:18) n − m (cid:19) q (cid:18) q k + ij (cid:19) q = b m +1 (cid:18) nm + 1 (cid:19) q (cid:18) q k + ij (cid:19) q , i < j ; u qn + m ( x ) = w m ( x ) u n ( x q ) + qb n x m +1 w q − − m ( x ) u n − ( x q ) ,g qn + m ( x ) = x m w q − − m ( x ) g n ( x q ) + qb n +1 w m − ( x ) g n +1 ( x q ) , ≤ m < q. According to the definition of the matrix [ q ] P , P = [2] P × [3] P × [5] P × ... × [ p ] P × ..., where p is a member of the sequence of prime numbers. Respectively, the exponentialseries is the Hadamard product of the series ∞ Y n =0 w p − (cid:16) x p n /p pn − p − (cid:17) . Generalization of the matrix [ q ] P is the matrix [ ϕ,q ] P = ϕ,q P × ϕ,q P × ϕ,q P × ... × ϕ,q k P × ..., [ q,q ] P = [ q ] P . If ϕ = 0 , identities for the matrix [ ϕ,q ] P can be obtained from the identitiesfor the matrix [ q ] P by replacing b q k n = q k b n , c qn + i = c n /q n on b q k n = ϕ k b n , c qn + i = c n /ϕ n .If q is fixed, matrices [ ϕ,q ] P form the group: [ ϕ,q ] P × [ β,q ] P = [ ϕβ,q ] P, where [1 ,q ] P = ,q P = P (1 − x ) − is the identity element of the group of generalized Pascalmatrices. If ϕ = 0 , we get zero generalized Pascal matrix [0 ,q ] P = ,q P × ,q P × ,q P × ... × ,q k P × ... For example (Pascal triangle modulo 2), [0 , P = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . (cid:0) [0 ,q ] P (cid:1) n,m = 1 , n (cid:0) mod q k (cid:1) ≥ m (cid:0) mod q k (cid:1) ; = 0 , n (cid:0) mod q k (cid:1) < m (cid:0) mod q k (cid:1) , k = 1 , , . . . Fractal zero generalized Pascal matrices
Denote (cid:0) [0 ,q ] P (cid:1) n,m = (cid:18) nm (cid:19) ,q . Theorem. (cid:18) q k n + iq k m + j (cid:19) ,q = (cid:18) nm (cid:19) ,q (cid:18) ij (cid:19) ,q , ≤ i, j < q k . (3) Proof.
By definition, if n (cid:0) mod q k (cid:1) < m (cid:0) mod q k (cid:1) for some value of k , then (cid:0) nm (cid:1) ,q = 0 .We represent the numbers n , m in the form n = ∞ X i =0 n i q i , m = ∞ X i =0 m i q i , ≤ n i , m i < q. Then n (cid:0) mod q k (cid:1) = k − X i =0 n i q i , m (cid:0) mod q k (cid:1) = k − X i =0 m i q i . If n (cid:0) mod q k (cid:1) < m (cid:0) mod q k (cid:1) , then n i < m i at least for one i . Since (cid:0) nm (cid:1) ,q = 1 , if m i ≥ n i ,then true the identity (cid:18) nm (cid:19) ,q = ∞ Y i =0 (cid:18) n i m i (cid:19) ,q . (4) It remains to note that if n = ∞ X i =0 n i q i = q k s + j, ≤ j < q k , then j = k − X i =0 n i q i , s = ∞ X i =0 n i + k q i . In the works [7], [8], [9] matrices [0 ,q ] P are called generalized Sierpinski matrices. Theproperty (3) is represented as S q = S q,k ⊗ S q,k ⊗ S q,k ⊗ ..., k = 1 , , . . . , where S q = [0 ,q ] P , S q,k is the matrix consisting of q k first rows of the matrix S q , ⊗ denotesthe Kronecker multiplication. Matrices S q have a generalizations, one of which can berepresented as S q ( a ( x )) = S q,k ( a ( x )) ⊗ S q,k ( a ( x )) ⊗ S q,k ( a ( x )) ⊗ ..., where S q, ( a ( x )) is the matrix consisting of q first rows of the matrix A : [ ↑ , n ] A = x n a ( x ) , ( A ) n,m = a n − m ,S q,k ( a ( x )) is the matrix consisting of q k first rows of the matrix S q ( a ( x )) . Then S q ( a ( x )) S q ( b ( x )) = S q ( a ( x ) b ( x )) . The results of these works have been developed in [10], [11]. Introduced matrices T ( q ) = T ( q ) k ⊗ T ( q ) k ⊗ T ( q ) k ⊗ ..., k = 1 , , . . . , T ( q )1 is the matrix consisting of q first rows of the Pascal matrix, T ( q ) k is the matrixconsisting of q k first rows of the matrix T ( q ) . For example, T (3) = . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... . . . . Denote (cid:0) T ( q ) (cid:1) n,m = (cid:18) nm (cid:19) q . Then (cid:18) nm (cid:19) q = ∞ Y i =0 (cid:18) n i m i (cid:19) , n = ∞ X i =0 n i q i , m = ∞ X i =0 m i q i , ≤ n i , m i < q, n X m =0 (cid:18) nm (cid:19) q x m = ∞ Y i =0 (cid:16) x q i (cid:17) n i . These results become more transparent if we use algebra of the matrices ( a ( x ) | q ) : (( a ( x ) | q )) n,m = a n − m (cid:18) nm (cid:19) ,q , ( a ( x ) | q ) ( b ( x ) | q ) = ( a ( x ) ◦ b ( x ) | q ) , [ x n ] a ( x ) ◦ b ( x ) = n X m =0 (cid:18) nm (cid:19) ,q a m b n − m . Denote q k − X n =0 b n x n a (cid:16) x q k (cid:17) | q = ( a ( x ) , b ( x ) | q, k ) . For example, ( a ( x ) , b ( x ) | ,
1) = a b . . .a b a b . . .a b a b . . .a b a b a b a b . . .a b a b . . .a b a b a b a b . . .a b a b a b a b . . .a b a b a b a b a b a b a b a b . . . ... ... ... ... ... ... ... ... . . . , a ( x ) , b ( x ) | ,
2) = a b . . .a b a b . . .a b a b . . .a b a b a b a b . . .a b a b . . .a b a b a b a b . . .a b a b a b a b . . .a b a b a b a b a b a b a b a b . . . ... ... ... ... ... ... ... ... . . . . Since h x q k n + i i q k − X n =0 b n x n a (cid:16) x q k (cid:17) = a n b i , ≤ i < q k , then (cid:0) q k n + i, q k m + j (cid:1) -th element of the matrix ( a ( x ) , b ( x ) | q, k ) is equal to a n − m b i − j (cid:18) q k n + iq k m + j (cid:19) ,q = a n − m (cid:18) nm (cid:19) ,q b i − j (cid:18) ij (cid:19) ,q , ≤ i, j < q k . Thus, ( a ( x ) , b ( x ) | q, k ) is the block matrix, ( n, m ) -th block of which is the matrix a n − m (cid:18) nm (cid:19) ,q ( b ( x ) | q ) q k , where ( b ( x ) | q ) q k is the matrix consisting of q k first rows of the matrix ( b ( x ) | q ) . Hence, ( a ( x ) , b ( x ) | q, k ) ( c ( x ) , d ( x ) | q, k ) = ( a ( x ) ◦ c ( x ) , b ( x ) ◦ d ( x ) | q, k ) . If ( a ( x ) | q ) = ( a ( x ) , a ( x ) | q, , as in the case of the matrix [0 ,q ] P , then ( a ( x ) | q ) = ( a ( x ) , a ( x ) | q, k ) for all k = 1 , , . . . : a ( x ) = q − X n =0 a n x n ! a ( x q ) = q k − X n =0 a n x n a (cid:16) x q k (cid:17) = ∞ Y m =0 q k − X n =0 a n x nq mk ,a = 1 , a q k n + i = a n a i , ≤ i < q k , (( a ( x ) | q )) q k n + i,q k m + j = (( a ( x ) | q )) n,m (( a ( x ) | q )) i,j . (5) Hence a n = ∞ Y i =0 a n i , n = ∞ X i =0 n i q i = n + q ( n + q ( n + ... )) , ≤ n i < q. In view of the identity (4), a n − m (cid:18) nm (cid:19) ,q = ∞ Y i =0 a n i − m i , n = ∞ X i =0 n i q i , m = ∞ X i =0 m i q i , if n i ≥ m i for all i , and a n − m (cid:0) nm (cid:1) ,q = 0 in other case. I.e. (( a ( x ) | q )) n,m = ∞ Y i =0 (( a ( x ) | q )) n i ,m i . (6) ( a ( x ) | q ) , denote them u n ( x ) , form therecurrent sequences: u n ( x ) = n X m =0 a n − m x m , ≤ n < q ; u qn + i ( x ) = u n ( x q ) u i ( x ) , ≤ i < q ; or u n ( x ) = ∞ Y i =0 u n i (cid:16) x q i (cid:17) , n = ∞ X i =0 n i q i , ≤ n i < q. If ( a ( x ) | q ) ( b ( x ) | q ) = ( a ( x ) ◦ b ( x ) | q ) , where a ( x ) = q − X n =0 a n x n ! a ( x q ) , b ( x ) = q − X n =0 b n x n ! b ( x q ) , then [ x n ] a ( x ) ◦ b ( x ) = ∞ Y i =0 c n i , c n i = [ x n i ] a ( x ) b ( x ) , n = ∞ X i =0 n i q i , ≤ n i < q. For example, [0 , P = (cid:18) − x | (cid:19) = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 8 8 4 8 4 4 2 8 4 4 2 4 2 2 1 . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . , [ x n ] (1 − x ) − ◦ (1 − x ) − = ∞ Y i =0 n i , n = ∞ X i =0 n i i , ≤ n i < , [ n, → ] [0 , P = ∞ Y i =0 (cid:16) x i (cid:17) n i . Let c ( x ) = q − X n =0 c n x n ! c ( x q ) . Then (cid:0) P c ( x ) (cid:1) q k n + i,q k m + j = c q k n + i c q k ( n − m )+ i − j c q k m + j = c n c i c n − m c i − j c m c j = (cid:0) P c ( x ) (cid:1) n,m (cid:0) P c ( x ) (cid:1) i,j , ≤ i, j < q k , i ≥ j . Hence, matrices P c ( x ) × [0 ,q ] P and ( c ( x ) | q ) have the same fractalproperties. Applied to the matrix T ( q ) : T ( q ) = P c ( x ) × [0 ,q ] P, c n = ( n !) − , ≤ n < q. eferences [1] G. Fontene, Generalization d’une formule connue, Nouv. ann. math., 1915, 15 (4), p.112.[2] S. M. Roman, The Umbral Calculus, Academic Press, 1984.[3] C. T. Long, Some Divisibility Properties of Pascal’s Triangle, The Fibonacci Quar-terly, 1981, Vol. 19, № 3, p. 257-263.[4] С. К. Абачиев, О треугольнике Паскаля, простых делителях и фрактальныхструктурах, В мире науки, 1989, № 9, с. 75-78.[5] B.A. Bondarenko, Generalized Pascal triangles and pyramids, their fractals, graphsand applications, Santa Clara, CA : Fibonacci Association, 1993.[6] Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized BinomialCoefficients, and Kummer’s Theorem, Mathematics Magazine, Vol. 87, No. 2, April2014, p. 135-143.[7] Hieu D. Nguyen, A Generalization of the Digital Binomial Theorem, Journal ofInteger Sequences, Vol. 18, 15.5.7, 2015.[8] Toufik Mansour, Hieu D. Nguyen, A q -Digital Binomial Theorem, arXiv: 1506.07945.[9] Toufik Mansour, Hieu D. Nguyen, A Digital Binomial Theorem for Sheffer Sequences,arXiv:1510.08529.[10] Lin Jiu, Christophe Vignat, On Binomial Identities in Arbitrary Bases, Journal ofInteger Sequences, Vol. 19, 16.5.5, 2016.[11] Tanay Wakhare, Christophe Vignat, Base- bb