Riordan-Dirichlet Group
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∞ ϕn ∞ s (ϕ) aϕ (x) = exp (ϕ log a (x)) = (log a (x))n = n xn, n! n! n=0 n=0 X X where sn (ϕ)/n! are the polynomials in ϕ of degree≤ n called convolution polynomials [7]. Explicit form of these polynomials:
s (ϕ) n bm1 bm2 ... bmn s (ϕ)=1, n = ϕm 1 2 n , 0 n! m !m ! ... m ! m=1 1 2 n X X i m where bi = [x ] log a (x) and summation of the coefficient of ϕ is over all monomials m1 m2 mn n n b1 b2 ... bn for which i=1 imi = n, i=1 mi = m. Matrix of the differential operator in the space of formal power series will be denoted P P by Dx:
∞ ′ n−1 Dxa (x)= a (x)= nanx . n=1 X Then
(a (x) b (x))′ = a (x) b′ (x)+ a′ (x) b (x) , (an (x))′ = nan−1 (x) a′ (x) ,
∞ ϕ − 1 (aϕ (x))′ = ϕa′ (x) (a (x) − 1)n−1 = ϕaϕ−1 (x) a′ (x) , n − 1 n=1 X ∞ (log a (x))′ = a′ (x) (−1)n−1(a (x) − 1)n−1 = a′ (x) a−1 (x) . n=1 X Theorem 1. Each formal power series a (x), a0 = 1, is associated by means of the transform ∞ xn aϕ (x)= [xn] 1 − xβ(log a (x))′ aϕ+βn (x) (1) aβn (x) n=0 X with the set of series (β)a (x), (0)a (x)= a (x), such that
−β β (β)a xa (x) = a (x) , a x(β)a (x) = (β)a (x) , ϕ [xn] aϕ (x) = [xn] 1 − xβ(log a (x))′ aϕ+βn (x)= [xn] aϕ+βn (x) , (β) ϕ + βn
′ ϕ + βn [xn] 1+ xβ log a (x) aϕ (x)= [xn] aϕ (x) = [xn] aϕ+βn (x) . (β) (β) ϕ (β) Remark 2. Transformation (1) is a particular case of the Lagrange series expansion. But, considering our plans, we give an alternative proof which relies only on the properties of the Riordan matrices. −1 Proof. If the matrices (1, a (x)), a0 =1, (1, b (x)), b0 =1, are mutually inverse, then
1, a−1 (x) b (x)= a (x) , (1, b (x)) a (x)= b (x) .
Since (xnbn (x))′ = nxn−1bn−1 (x) b (x) 1+ x(log b (x))′ , then ′ Dx (1, b (x)) = b (x) 1+ x(log b (x)) , b (x) Dx, 3 (log b (x))′ (1, b (x)) a′ (x)= . 1+ x(log b (x))′ From this we find: −1 1+ x(log b (x))′, b (x) = 1 − x(log a (x))′, a−1 (x) . Denote n m (m) n ′ m (m) [x ] a (x)= an , [x ] 1 − x(log a (x)) a (x)= cn , ∞ ∞ (m+n) n (m+n) n am (x)= an x , cm (x)= cn x . n=0 n=0 X X m We construct the matrix A mth column of which has the generating function x am (x) m and the matrix C mth column of which has the generating function x cm (x): (0) (0) a0 0 0 0 ··· c0 0 0 0 ··· (1) (1) (1) (1) a1 a0 0 0 ··· c1 c0 0 0 ··· A = a(2) a(2) a(2) 0 ··· , C = c(2) c(2) c(2) 0 ··· . 2 1 0 2 1 0 a(3) a(3) a(3) a(3) ··· c(3) c(3) c(3) c(3) ··· 3 2 1 0 3 2 1 0 . . . . . . . . . . ...... ...... It’s obvious that [n, →] A = [n, →](an (x) , 1) , [n, →] C = [n, →] 1 − x(log a (x))′ an (x) , 1 . Since x 1 − xa′ (x) a−1 (x) am (x)= am (x) − (am (x))′, m or m − n [xn] 1 − x(log a (x))′ am (x)= [xn] am (x) , m then xn+m Axm 1 − x(log a (x))′ a−m (x)= xn+m Cxma−m (x)= n ′ n = [x ] 1 − x(log a (x)) a (x)=1, n=0; =0,n> 0. Thus, A = 1+ x(log b (x))′, b (x) , C = (1, b (x)) , m + n [xn] 1+ x(log b (x))′ bm (x)= [xn] bm (x) = [xn] am+n (x) , m m [xn] bm (x) = [xn] 1 −x(log a (x))′ am+n (x)= [xn] am+n (x) . m + n Denote −β −1 β 1, a (x) = 1, (β)a (x) . Then βm [xn] aβm (x)= [xn] aβm+βn (x) . (β) βm + βn
Let sn (ϕ)/n! are the convolution polynomials of the series a (x). Then ∞ ϕ s (ϕ + βn) aϕ (x)= n xn. (β) ϕ + βn n! n=0 X We note that the identity −1 [n, →] 1 − x(log a (x))′, a−1 (x) = [n, →](an (x) , 1) is equivalent to the Lagrange expansion formula for an arbitrary series f (x): f (x) ∞ xn = [xn] f (x) an (x) . ′ an (x) 1 − x(log a (x)) n=0 X 4 3 Riordan-Dirichlet matrices
In this section, we construct the matrices of certain operators in the space of formal Dirichlet series. We will associate the columns of matrices with the generating functions of their elements, i.e. with the formal Dirichlet series. Numbering of the rows and columns −s −s begins with 1. Coefficient operator will be denoted by [n ]: [n ] a (s)= an. Matrix nth column of which has the generating function n−sa (s) will be denoted by ha (s) , 1i: a1 0000000 ··· a a 000000 ··· 2 1 a3 0 a1 00000 ··· a a 0 a 0 0 0 0 ··· 4 2 1 a 0 0 0 a 0 0 0 ··· ha (s) , 1i = 5 1 . a a a 0 0 a 0 0 ··· 6 3 2 1 a 00000 a 0 ··· 7 1 a a 0 a 0 0 0 a ··· 8 4 2 1 ...... ...... Then ∞ ∞ n −s −s ha (s) , 1i b (s)= bnn a (s)= n bdan/d = a (s) b (s) , n=1 n=1 X X Xd|n where symbol d|n means that the summation is over all natural divisors d of the number n. Thus, matrix ha (s) , 1i corresponds to the operator of multiplication by the series a (s). Set of all such matrices form an algebra isomorphic to the algebra of formal Dirichlet series: ha (s) , 1i + hb (s) , 1i = ha (s)+ b (s) , 1i , ha (s) , 1ihb (s) , 1i = hb (s) , 1iha (s) , 1i = ha (s) b (s) , 1i .
Remark 3. Matrices ha (s) , 1i, denoted by A = [(an)] , were introduced in [8], [9]. In [9], [10], they are considered as matrices of the operators in the Hilbert space and are called (in [10]) the Dirichlet matrices, or D-matrices. In the algebra of formal Dirichlet series, the power and logarithm of the series a (s), a1 =1, are defined as
∞ ∞ ϕ (−1)n−1 aϕ (s)= (a (s) − 1)n, log a (s)= (a (s) − 1)n. n n n=0 n=1 X X Power of the series can also be defined as
∞ ϕn ∞ aϕ (s) = exp (ϕ log a (s)) = (log a (s))n = h (ϕ)n−s, n! n n=0 n=1 X X where hn (ϕ) are the polynomials in ϕ of degree < n which, like in the case of power series, can be called convolution polynomials. Explicit form of these polynomials:
n bm2 bm3 ... bmn h (ϕ)=1, h (ϕ)= ϕm 2 3 n , 1 n m !m ! ... m ! m=1 2 3 n X X i m where bi = [x ] log a (s) and summation of the coefficient of ϕ is over all monomials m2 m3 mn n mi n b2 b3 ... bn for which i=2 i = n, i=2 mi = m. Q P
5 Matrix nth column of which has the generating function n−saln n (s) will be denoted by h1, a (s)i:
10000000 ··· (2) 0 a1 000000 ··· (3) 0 0 a1 0 0 0 0 0 ··· (2) (4) 0 a 0 a 0 0 0 0 ··· 2 1 (5) h1, a (s)i = 00 0 0 a1 0 0 0 ··· , (2) (3) (6) 0 a3 a2 0 0 a1 0 0 ··· (7) 00 0 0 0 0 a 0 ··· 1 0 a(2) 0 a(4) 0 0 0 a(8) ··· 4 2 1 ...... ...... where −s ln m (m) (1) (1) n a (s)= an , a1 =1, an =0. Then ∞ ∞ n −s ln n −s (d) h1, a (s)i b (s)= bnn a (s)= n bdan/d = b (s) ◦ a (s) , n=1 n=1 X X Xd|n where sign ◦ means the operation under consideration similar to a composition of power series. Denote hb (s) , 1ih1, a (s)i = hb (s) , a (s)i .
Theorem 2. Matrices hb (s) , a (s)i, b1 =6 0, a1 =6 0, form a group with respect to the matrix multiplication whose elements are multiplied by the rule:
hb (s) , a (s)ihf (s) ,g (s)i = hb (s)(f (s) ◦ a (s)) , a (x)(g (s) ◦ a (s))i .
Proof. Since
∞ ∞ −s −s ln mn −s ln m −s ln n h1, a (s)i m b (s)= bn(mn) a (s)= m a (s) bnn a (s) , n=1 n=1 X X then h1, a (s)ihb (s) , 1i = hb (s) ◦ a (s) , a (s)i . Hence, h1, a (s)i b (s) c (s)=(b (s) ◦ a (s))(c (s) ◦ a (s)) , h1, a (s)i bn (s)=(b (s) ◦ a (s))n, and by definition of power of series
h1, a (s)i bϕ (s)=(b (s) ◦ a (s))ϕ.
Then h1, a (s)i m−sbln m (s)= m−saln m (s)(b (x) ◦ a (x))ln m, h1, a (s)ih1, b (s)i = h1, a (s)(b (s) ◦ a (s))i . If h1, a (s)i−1b−1 (s)= f (s) , h1, a (s)i−1a−1 (s)= g (s) , then hb (s) , a (s)i−1 = hf (s) ,g (s)i .
6 Matrix of the differential operator in the space of formal Dirichlet series will be denoted by Ds: ∞ ′ −s Dsa (s)= a (s)= ln (1/n ) ann . n=1 X Then (a (s) b (s))′ = a (s) b′ (s)+ a′ (s) b (s) , (an (s))′ = nan−1 (s) a′ (s) , ∞ ϕ − 1 (aϕ (s))′ = ϕa′ (s) (a (s) − 1)n−1 = ϕaϕ−1 (s) a′ (s) , n − 1 n=1 X∞ (log a (s))′ = a′ (s) (−1)n−1(a (s) − 1)n−1 = a′ (s) a−1 (s) . n=1 X Theorem 3. Each formal Dirichlet series a (s), a1 = 1, is associated by means of the transform ∞ n−s aϕ (s)= n−s 1+ β(log a (s))′ aϕ+β ln n (s) aβ ln n (s) n=1 X with the set of series (β)a (s), (0)a (s)= a (s), such that ϕ n−s aϕ (s)= n−s 1+ β(log a (s))′ aϕ+β ln n (s)= n−s aϕ+β ln n (s) , (β) ϕ + β ln n ϕ+ β ln n n−s 1 − β log a (s) ′ aϕ (s)= n−s aϕ (s)= n−s aϕ+β ln n (s) . (β) (β) ϕ (β) −1 Proof. If the matrices h1, a (s)i, a1 =1, h1, b (s)i, b1 =1, are mutually inverse, then 1, a−1 (s) b (s)= a (s) , h1, b (s)i a (s)= b (s) . Since ′ n−sbln n (s) = n−sbln n (s) ln (1/n ) 1 − b′ (s) b−1 (s) , then ′ Ds h1, b (s)i = 1 − (log b (s)) , b (s) Ds, b′ (s) h1, b (s)i a′ (s)= . 1 − (log b (s))′ From this we find: −1 1 − (log b (s))′, b (s) = 1 + (log a (s))′, a−1 (s) . Denote −s ln m (m) −s ′ ln m (m) n a (s)= an , n 1 + (log a (s)) a (s)= cn , ∞ ∞ (mn) −s (mn) −s am (s)= an n , cm (s)= cn n . n=1 n=1 X X −s We construct the matrix A mth column of which has the generating function m am (s) −s and the matrix C mth column of which has the generating function m cm (s):
(1) (1) a1 0 0 0 0 0 ··· c1 0 0 0 0 0 ··· (2) (2) (2) (2) a2 a1 0 0 0 0 ··· c2 c1 0 0 0 0 ··· (3) (3) (3) (3) a3 0 a1 0 0 0 ··· c3 0 c1 0 0 0 ··· (4) (4) (4) (4) (4) (4) a a 0 a 0 0 ··· , c c 0 c 0 0 ··· 4 2 1 4 2 1 a(5) 0 0 0 a(5) 0 ··· c(5) 0 0 0 c(5) 0 ··· 5 1 5 1 (6) (6) (6) (6) (6) (6) (6) (6) a6 a3 a2 0 0 a1 ··· c6 c3 c2 0 0 c1 ··· ...... ...... ...... ...... 7 It’s obvious that [n, →] A = [n, →] aln n (s) , 1 , [n, →] C = [n, →] 1 + (log a (s))′aln n (s) , 1 . Since 1 ′ 1+ a′ (s) a−1 (s) aln m (s)= aln m (s)+ aln m (s) , ln m or ln (m/n ) n−s 1 + (log a (s))′ aln m (s)= n−s aln m (s) , ln m then
(nm)−s Am−s 1 + (log a (s))′ a− ln m (s)= (nm)−s Cm−sa− ln m (s)=
= n−s 1 + (log a (x))′ aln n (s)=1, n=1; =0,n> 1. Thus, A = 1 − (log b (s))′, b (s) , C = h1, b (s)i , ln mn n−s 1 − (log b ( s))′ bln m (s)= n−s bln m (s)= n−s aln mn (s) , ln m ln m n−s bln m (s)= n−s 1 + (log a (s))′ aln mn (s)= n−s aln mn (s) . ln mn Denote −β −1 β 1, a (s) = 1, (β)a (s) . Then β ln m n−s aβ ln m (s)= n−s aβ ln m+β ln n (s) . (β) β ln m + β ln n Let hn (ϕ) are the convolution polynomials of the series a (s). Then
∞ ϕ aϕ (s)= h (ϕ + β ln n) n−s. (β) ϕ + β ln n n n=1 X We note that the identity
−1 [n, →] 1 + (log a (s))′, a−1 (s) = [n, →] aln n (s) , 1 is equivalent to a formula similar to the Lagrange expansion formula:
f (s) ∞ n−s = n−s f (s) aln n (s) . ′ aln n (s) 1 + (log a (s)) n=1 X 4 Some examples
In this section, we show how matrices h1, a (s)i can be used for the next construction. Each formal power series a (x), a0 =1, with convolution polynomials sn (ϕ)/n! is put in correspondence with the formal Dirichlet series a (s) such that
∞ ∞ ∞ s (ϕ) s (ϕ) s (ϕ) ...s r (ϕ) aϕ (s)= n p−ns =1+ m1 m2 m n−s, n! m !m ! ... m ! p=2 n=0 ! n=2 1 2 r Y X X m1 m2 mr where the product is taken over all prime numbers and n = p1 p2 ... pr is the canon- ical decomposition of number n. With this correspondence, the group of series a (x) is isomorphic to the group of series a (s): if a (x) b (x) = c (x), then a (s) b (s) = c (s). If
8 m −s m [x ] log a (x)= bm, then [n ] log a (s)= bm when n = p , where p is the prime number, and [n−s] log a (s)=0 otherwise. Polynomials
sm1 (ϕ) sm2 (ϕ) ...smr (ϕ) hn (ϕ)= m1!m2! ... mr! are the convolution polynomials of the series a (s). Dirichlet series corresponding to the exponential series will be denoted by ε (s):
∞ ϕm1+m2+...+mr εϕ (s)=1+ n−s. m !m ! ... m ! n=2 1 2 r X −s Note that log ε (s)= p , where the summation is over all prime numbers. Let (1)ε (s) denote the series associated with ε (s) by Theorem 3. Then P ϕv(n) ϕ(ϕ + ln n)v(n)−1 n−s εϕ (s)= , n−s εϕ (s)= , f (n) (1) f (n) where
v (1) = 0, v (n)= m1 + m2 + ... + mr, f (1) = 1, f (n)= m1!m2!...mr!.
From ϕ+β ϕ β (1)ε (s)= (1)ε (s) (1)ε (s) we obtain an analog of the Abel’s generalized binomial formula:
n (ϕ + β)(ϕ + β + ln n)v(n)−1 = ϕ(ϕ + ln d)v(d)−1β(β + ln (n/d ))v(n/d )−1, d f Xd|n where n f (n) = . d f (d) f (n/d ) f Since −s ′ ϕ −s ϕ+ln n n 1 − log (1)ε (s) (1)ε (s)= n ε (s) , from ′ ϕ+β ′ ϕ β 1 − log (1)ε (s) (1)ε (s)= 1 − log (1)ε (s) (1)ε (s) (1)ε (s) we obtain n (ϕ + β + ln n)v(n) = (ϕ + ln d)v(d)β(β + ln (n/d ))v(n/d )−1. d f Xd|n Since n −s (d) (d) −s ln d n h1, a (s)i b (s)= bdan/d , an/d = (n/d ) a (s) , d|n X from ϕ ϕ ϕ −1 ϕ (1)ε (s)= 1, (1)ε (s) ε (s) , ε (x)= 1,ε (s) (1)ε (s) we obtain n ϕ(ϕ + ln n)v(n)−1 = ϕv(d) ln d(ln n)v(n/d )−1, d f Xd|n n ϕv(n) = ϕ(ϕ + ln d)v(d)−1(ln (1/d ))v(n/d ). d f Xd|n 9 When n = pm, where p is the prime number, obtained formulas take the form of Abel identities [3], [11, pp. 92-99]:
m m (ϕ + β)(ϕ + β + ma)m−1 = ϕ(ϕ + ka)k−1β(β +(m − k) a)m−k−1, k Xk=0 m m (ϕ + β + ma)m = (ϕ + ka)kβ(β +(m − k) a)m−k−1, k Xk=0 m m ϕ(ϕ + ma)m−1 = ϕkka(ma)m−k−1, k Xk=0 m m ϕm = ϕ(ϕ + ka)k−1(−ka)m−k, a = ln p. k Xk=0 We generalize this example. Let there be given the series a (s) such that
u (ϕ) n−s aϕ (s)= n , f (n) where m1 m2 mr un (ϕ)= sm1 (ϕ) sm2 (ϕ) ...smr (ϕ) , n = p1 p2 ... pr . Then ϕ u (ϕ + β ln n) n−s aϕ (s)= n , (β) ϕ + β ln n f (n) ϕ β ϕ ϕ −β ϕ (β)a (s)= 1, (β)a (s) a (s) , a (s)= 1, a (s) (β)a (s) , ϕ n ln d un (ϕ + β ln n)= ud (ϕ) un/d (β ln n) , ϕ + β ln n d f ln n Xd|n n ϕ un (ϕ)= ud (ϕ + β ln d)un/d (β ln (1/d )) . d f ϕ + β ln d Xd|n m Since upm (ϕ)= sm (ϕ), when n = p formulas take the form of mutually inverse relations for the Lagrange series:
m ϕ m k s (ϕ + ma)= s (ϕ) s (ma) , ϕ + ma m k k m m−k Xk=0 m m ϕ s (ϕ)= s (ϕ + ka) s (−ka) , a = β ln p. m k ϕ + ka k m−k Xk=0 n We note the identities for the coefficients d f similar to the identities
n n n n =2n; k(−1)n−k =0, n =16 . k k Xk=0 Xk=0 Since ε (s) ε (s)= ε2 (s); ε′ (x) ε−1 (x) = (log ε (x))′, then
n n =2v(n); ln d(−1)v(n/d ) =0, n =6 p. d f d f Xd|n Xd|n
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