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 Pascal matrices, based on which we are building fractal generalized Pascal matrices, is in- troduced. Pascal matrix (Pascal triangle) is the Hadamard product of the fractal generalized Pascal matrices whose elements equal to pk, where p is a fixed prime number, k = 0, 1, 2,... The concept of zero generalized Pascal matrices, an example of which is the Pascal triangle modulo 2, arise in connection with the system of matrices introduced. 1 Introduction Consider the following generalization of the binomial coefficients [1]. For the coefficients of the formal power series b (x), b0 =0; bn =06 , n> 0, denote n n bn! n b0!=1, bn!= bm, = ; =0, m > n. m b !b − ! m m=1 b m n m b Y Then n n − 1 b − b n − 1 = + n m . m m − 1 b − m b b n m b n-th coefficient of the series a (x), (n, m)-th element of the matrix A, n-th row and n-th column of the matrixA will be denoted respectively by n [x ] 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 arXiv:1612.00970v1 [math.NT] 3 Dec 2016 c0c0 0 0 0 0 ... c0 c0c1 c1c0 0 0 0 ... c1 c1 c0c2 c1c1 c2c0 0 0 ... c2 c2 c2 Pc(x) = c0c3 c1c2 c2c1 c3c0 0 ... c3 c3 c3 c3 c0c4 c1c3 c2c2 c3c1 c4c0 ... c4 c4 c4 c4 c4 . . .. cmcn−m R Pc(x) n,m = , cn ∈ , cn =06 . cn Denote [↑, 1]Pc(x) = b (x). If c0 =1, then cn n c = 1 , P = . n b ! c(x) n,m m n b 1 Let c0 =1. Since Pc(x) = Pc(ϕx), we take for uniqueness that c1 =1. Matrix Pc(x) will be called generalized Pascal matrix. If c (x)= ex, it turns into Pascal matrix 10000 ... 11000 ... 12100 ... P = 13310 ... 14641 ... . . .. In Sec. 2. we consider the set of generalized Pascal matrices as a group under Hadamard multiplication and introduce a special system of matrices, which implies the concept of zero generalized Pascal matrices; as an example, we consider the matrix of the q-binomial coefficients for q = −1. In Sec. 3. we construct a fractal generalized Pascal matrices whose Hadamard product is the Pascal matrix. In Sec. 4. we consider the fractal zero generalized Pascal matrices that were previously considered in [7] – [11] (generalized Sierpinski matrices and others). 2 Special system of generalized Pascal matrices − Elements of the matrix Pc(x), – denote them Pc(x) n,m = cmcn m/cn =(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, 1, 2, . It means that each matrix Pc(x) can be associated with the algebra of formal power series whose elements are multiplied by the rule n a (x) ◦ b (x)= g (x) , gn = (n, m) ambm−n, m=0 X that is, if (A)n,m = an−m (n, m), (B)n,m = bn−m (n, m), (G)n,m = gn−m (n, m), then AB = BA = G: n −1 gn =(n + p,p) (n + p, m + p)(m + p,p) an−mbm = m=0 X n −1 =(n + q, q) (n + q, m + q)(m + q, q) an−mbm. m=0 X The set of generalized Pascal matrix is a group under Hadamard multiplication (we denote this operation ×): ∞ n Pc(x) × Pg(x) = Pc(x)×g(x), c (x) × g (x)= cngnx . n=0 X Introduce the special system of matrices q−1 − xq 1 P = P (ϕ)= P , c (ϕ,q,x)= xn 1 − , q> 1. ϕ,q q c(ϕ,q,x) ϕ n=0 ! X 2 Then 1 1 c = , 0 ≤ i < q; c − = , 0 < i ≤ q, qn+i ϕn qn i ϕn−1 n n cqm+jcq(n−m)+i−j ϕ ϕ = m n−m =1, i ≥ j; = m n−m−1 = ϕ, i < j, cqn+i ϕ ϕ ϕ ϕ or (ϕ,qP )n,m =1, n (modq) ≥ m (modq); = ϕ, n (modq) < m (modq) . For example, ϕ,2P , ϕ,3P : 100000000 ... 100000000 ... 110000000 ... 110000000 ... 1 ϕ 1000000 ... 111000000 ... 111100000 ... 1 ϕ ϕ 100000 ... 1 ϕ 1 ϕ 10000 ... 1 1 ϕ 110000 ... 111111000 ... , 111111000 ... 1 ϕ 1 ϕ 1 ϕ 1 0 0 ... 1 ϕ ϕ 1 ϕ ϕ 1 0 0 ... 111111110 ... 1 1 ϕ 1 1 ϕ 1 1 0 ... 1 ϕ 1 ϕ 1 ϕ 1 ϕ 1 ... 111111111 ... . . . .. .. Elements of the matrix ϕ,qP × Pc(x) satisfy the identities (1), (2) for any values ϕ, so it makes sense to consider also the case ϕ =0 since it corresponds to a certain algebra of formal power series (which, obviously, contains zero divisors; for example, in the algebra 2 associated with the matrix 0,2P × Pc(x) the product of the series of the form xa (x ) is zero). It is clear that in this case the series c (ϕ,q,x) is not defined. Matrix 0,qP × Pc(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 Pg(q,x): − x n 1 P = [xn] = qm, q ∈ R. g(q,x) n,1 (1 − x) (1 − qx) m=0 X − Here g (0, x)=(1 − x) 1, g (1, x)= ex. In other cases (the q-umbral calculus [2]), except q = −1, ∞ (q − 1)n n g (q, x)= xn, (qn − 1)! = (qm − 1), q0 − 1 !=1. (qn − 1)! n=0 m=1 X Y −1 Matrices Pg(q,x), Pg(q,x) also can be defined as follows: n n−1 n m −1 −1 m [↑, n]Pg(q,x) = x (1 − q x) , [n, →]Pg(q,x) = (x − q ). m=0 m=0 Y Y −1 When q = −1 we get the matrices Pg(−1,x), Pg(−1,x): 1000000 ... 1 0 0 0 000 ... 1100000 ... −11 0 0 000 ... 1010000 ... −10 1 0 000 ... 1111000 ... 1 −1 −11 000 ... 1020100 ... , 1 0 −20 100 ... , 1122110 ... −1 1 2 −2 −1 1 0 ... 1030301 ... −10 3 0 −3 0 1 ... . . . .. . .. 3 where the series g (−1, x) is not defined. Since 1−j 2m+j 2n+i (1 + x) x n Pg(−1,x) = x = , i ≥ j; =0, i<j; 2n+i,2m+j 2 m+1 m (1 − x ) i, j =0,1, then x2 Pg(−1,x) = 0,2P × Pc(x), c (x)=(1+ x) e : 1 1 c = , 0 ≤ i< 2; c − = , 0 < i ≤ 2, 2n+i n! 2n i (n − 1)! n n − 1 P = , i ≥ j; = n , i < j. c(x) 2n+i,2m+j m m A generalization of this matrix is the matrix q−1 n xq 0,qP × Pc(x), c (x)= x e , n=0 ! X n n − 1 P = , i ≥ j; = n , i<j; 0 ≤ i, j < q. c(x) qn+i,qm+j m m Each nonzero generalized Pascal matrix is the Hadamard product of the matrices ϕ,qP . Since the first column of the matrix Pc(x), – denote it b (x), – is the Hadamard product of the first columns of the matrices ϕ,qP , – denote them ϕ,qb (x): n [x ] ϕ,qb (x)=1, n (modq) =0;6 = ϕ, n (modq)=0, so that Pc(x) = 2P (b2) × 3P (b3) × 4P (b4/b2 ) × 5P (b5) × 6P (b6/b2b3 ) × 7P (b7) × ×8P (b8/b4 ) × 9P (b9/b3 ) × 10P (b10/b2b5 ) × 11P (b11) × 12P (b12b2/b4b6 ) × ... and so on. Let eq is a basis vector of an infinite-dimensional vector space. Mapping of the set of generalized Pascal matrices in an infinite-dimensional vector space such that ϕ,qP → eq log |ϕ| is a group homomorphism whose kernel consists of all involutions in the group of generalized Pascal matrices, i.e. from matrices whose non-zero elements equal to ±1. Thus, the set of generalized Pascal matrices whose elements are non-negative numbers is an infinite-dimensional vector space. Zero generalized Pascal matrices can be viewed as points at infinity of space. 3 Fractal generalized Pascal matrices 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 binomial coefficients. We consider them from point of view based on the system of matrices ϕ,qP . We start with the matrix [2]P = 2,2P × 2,22 P × 2,23 P × ... × 2,2k P × ... 4 1000000000000000 ... 1100000000000000 ... 1210000000000000 ... 1111000000000000 ... 1424100000000000 ... 1122110000000000 ... 1214121000000000 ... 1111111100000000 ... 1848284810000000 ... [2]P = . 1144224411000000 ... 1218242812100000 ... 1111222211110000 ... 1424184814241000 ... 1122114411221100 ... 1214121812141210 ... 1111111111111111 ... . . .. The first column of the matrix [2]P , – denote it b (x), – is the Hadamard product of the series 2,2k b (x) , k =1, 2 ,... , n k k [x ] 2,2k b (x)=1, n mod2 =0;6 =2, n mod2 =0. It is the generating function of the distribution of divisors 2kin the series of natural numbers: b (x)= x +2x2 + x3 +4x4 + x5 +2x6 + x7 +8x8 + x9 +2x10 + x11 +4x12+ +x13 +2x14 + x15 + 16x16 + x17 +2x18 + x19 +4x20 + x21 +2x22 + x23 + ..