polynomials of multipartitional type and inverse relations

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University of Science and Technology Houari Boumediene, USTHB. Faculty of .... admits formal power series in x and let F〈−1〉 denote the compositional inverse ...
Discussiones Mathematicae General Algebra and Applications 31 (2011) 185–199

POLYNOMIALS OF MULTIPARTITIONAL TYPE AND INVERSE RELATIONS Miloud Mihoubi University of Science and Technology Houari Boumediene, USTHB Faculty of Mathematics, P.B. 32 El Alia, 16111, Algiers, Algeria e-mail: [email protected] or [email protected]

and Hac` ene Belbachir University of Science and Technology Houari Boumediene, USTHB Faculty of Mathematics, P.B. 32 El Alia, 16111, Algiers, Algeria e-mail: [email protected] or [email protected] Abstract Chou, Hsu and Shiue gave some applications of Fa` a di Bruno’s formula to characterize inverse relations. Our aim is to develop some inverse relations connected to the multipartitional type polynomials involving to binomial type sequences. Keywords: Bell polynomials, inverses relations, polynomials of multipartitional type, binomial type sequences. 2010 Mathematics Subject Classification: Primary 05A10; Secondary 11B83, 11B73, 40E15.

1.

Introduction

The partial (exponential) multipartitional polynomial Bn1 ,...,nr ;k (x0,...,0,1 , . . . , xn1 ,...,nr ) in the variables x0,...,0,1 , . . . , xn1 ,...,nr is defined by the sum (1)

Bn1 ,...,nr ,k (x0,...,0,1 , x0,...,0,2 , . . . , xn1 ,...,nr ) :=

P

n1 !···nr ! k0,...,0,1 !k0,...,0,2 !···kn1 ,...,nr !

x0,...,0,1 k0,...,0,1 0!···0!1!

···



xn1 ,...,nr n1 !···nr !

kn1 ,...,nr

,

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where the summation is extended over all partitions of the multipartite number (n1 , . . . , nr ) into k parts, that is, over all nonnegative integers k0,...,0,1 , . . . , kn1 ,...,nr solution of the equations n1X ,...,nr

(2)

ij ki1 ,...,ir = nj , j = 1, . . . , r,

i1 ,...,ir =0

n1X ,...,nr

ki1 ,...,ir = k

i1 ,...,ir =0

with convention k0,...,0 = 0. The exponential generating function for Bn1 ,...,nr ,k is given by X

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr )

n1 +···+nr ≥k

(3)

 1  = k!

X

tnr tn1 1 ··· r n1 ! nr !

k ti11 tirr  xi1 ,...,ir ··· . i1 ! ir !

i1 +···+ir ≥1

From the above definition, the partial multipartitional polynomials generalize the partial Bell polynomials introduced by Bell [3], see also [5, 9] and [10]. Chou et al. [4] gave some inverse relations related to Fa`a di Bruno’s formula. They have proved that any function F having power formal series with compositional inverse F h−1i , satisfy

yn =

n X

k Dx=a F (x) Bn,k (x1 , x2 , . . .) ,

xn =

n X

k h−1i Dx=F (x) Bn,k (y1 , y2 , . . .) . (a) F

k=1

k=1

In this paper, we show the role of the binomial type sequences to develop some inverse relations related to the multipartitional polynomials. In other words, we connect some inverse relations with the multipartitional polynomials and with some results given in [6] and [8] related to partial Bell polynomials.

2.

Multipartitional polynomials and inverse relations

Theorem 1. Let a be a real number and F be a function such that F (a + x) admits formal power series in x and let F h−1i denote the compositional inverse

Polynomials of multipartitional type and inverse relations

187

  of F so that F ◦ F h−1i (x) = F h−1i ◦ F (x) = x. Let ϕx (t1 , . . . , tr ) :=

X

xn1 ,...,nr

X

yn1 ,...,nr

n1 +···+nr ≥1

ϕy (t1 , . . . , tr ) :=

n1 +···+nr ≥1

tn1 1 tnr ··· r , n1 ! nr !

tn1 1 tnr ··· r . n1 ! nr !

Then we have the pair of inverse relations (4)

yn1 ,...,nr =

n1 +···+n X r

k Dx=a F (x)Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) ,

n1 +···+n X r

k h−1i Dx=F (x)Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) , (a) F

k=1

xn1 ,...,nr =

k=1

or equivalently (5)

F (a + ϕx (t1 , . . . , tr )) = F (a) + ϕy (t1 , . . . , tr ) .

Proof. The given conditions of the theorem ensure that there holds a pair of formal series expansions X

ϕy (t1 , . . . , tr ) =

n1 +···+nr ≥1

=

tn1 1 tnr ··· r n1 ! nr !

∞ X

k Dx=a F (x)

k=1

n1 +···+n X r

k Dx=a F (x)×

k=1

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) X tn1 1 tnr ··· r × n1 ! nr !

n1 +···+nr ≥k

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr )

=

∞ X

 1 k Dx=a F (x)  k!

∞ X

k Dx=a F (x)

k=1

=

k=1

X

k ti11 tirr  xi1 ,...,ir ··· i1 ! ir !

i1 +···+ir ≥1

(ϕx (t1 , . . . , tr ))k k!

= F (a + ϕx (t1 , . . . , tr )) − F (a) ,

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which states that ϕx (t1 , . . . , tr ) = F h−1i (F (a) + ϕy (t1 , . . . , tr )) − a ∞ X (ϕy (t1 , . . . , tr ))k k h−1i = Dx=F (a) F (x) , k! k=1

so, we have X

xn1 ,...,nr

n1 +···+nr ≥1

X

=

n1 +···+nr ≥1

tn1 1 tnr ··· r n1 ! nr !

tnr tn1 1 ··· r n1 ! nr !

n1 +···+n X r

k h−1i Dx=F (x) (a) F

k=1

× Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) . Then xn1 ,...,nr =

n1 +···+n P r k=1

k h−1i (x)B Dx=F n1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ). (a) F

Remark 1. Replacing xn1 ,...,nr by n1 ! · · · nr !xn1 ,...,nr and yn1 ,...,nr by n1 ! · · · nr !yn1 ,...,nr , relation (4) may be expressed in terms of the multivariate Fa`a di Bruno’s formula, as follows k F (x) X Dx=a yn1 ,...,nr = (x0,...,0,1 )k0,...,0,1 · · · (xn1 ,...,nr )kn1 ,...,nr , k0,...,0,1 ! · · · kn1 ,...,nr !

xn1 ,...,nr =

k h−1i (x) X Dx=F (a) F

k0,...,0,1 ! · · · kn1 ,...,nr !

(y0,...,0,1 )k0,...,0,1 · · · (yn1 ,...,nr )kn1 ,...,nr ,

where the summation is extended over all nonnegative integers k0,...,0,1 , . . . , kn1 ,...,nr solution of equations (2) and over k = 1, . . . , n1 + · · · + nr . Also, the pair of inverse relations given by relation (4) implies that for any sequence of real numbers (xn ) we have the following identity: n X

k h−1i Dx=F (x)Bn,k (y1 , . . . , yn ) = xn . (a) F

k=1

where yn =

j j=1 Dx=a F (x)Bn,j (x1 , . . . , xn ).

Pn

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189

Corollary 2. We have the pair of inverse relations (6)

yn1 ,...,nr =

n1 +···+n X r

Bn1 ,...,nr ,j (x0,...,0,1 , . . . , xn1 ,...,nr ) ,

xn1 ,...,nr =

n1 +···+n X r

(−1)j−1 (j − 1)!Bn1 ,...,nr ,j (y0,...,0,1 , . . . , yn1 ,...,nr ) .

j=1

j=1

Proof. It suffices to take in (4) F (x) = exp (x) and a = 0. Corollary 3. We have the pair of inverse relations (7)

yn1 ,...,nr = xn1 ,...,nr =

n1 +···+n X r k=1 n1 +···+n X r k=1

(α)k Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) ,   1 Bn ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) , α k 1

where (α)k = α (α − 1) · · · (α − k + 1) , k ≥ 1, and (α)0 = 1, α ∈ C. Proof. It suffices to take in (4) F (x) = xα and a = 1. Remark 2. For α = −1 in (7), we obtain (8)

yn1 ,...,nr = xn1 ,...,nr =

n1 +···+n X r

k=1 n1 +···+n X r

(−1)k k!Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , (−1)k k!Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) .

k=1

The inverse relations (8) are equivalent to    X X tn1 1 tnr r tn1 1 tnr r 1+ xn1 ,...,nr ··· 1+ yn1 ,...,nr ··· = 1, n1 ! nr ! n1 ! nr ! where the summations are taken over all nonnegative integers such that n1 +· · ·+ nr ≥ 1. Corollary 3 can be generalized by the following: Corollary 4. We have the pair of inverse relations n +···+nr   1 1X β αk Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , yn1 ,...,nr = β α k k=1   n1 +···+n 1 X r α xn1 ,...,nr = β k Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) . α β k k=1

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Proof. It suffices de take in (4) F (x) = a = 0.

(1+αt)β/α −1 , β

α, β ∈ C, αβ 6= 0, and

Theorem 5. Let {fn (x)} be a polynomial sequence of binomial type. Then we have the pair of inverse relations X fk−1 (a (k − 1) + x) (9) yn1 ,...,nr = Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , kx a (k − 1) + x X fk−1 (a (k − 1) − kx) Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) , xn1 ,...,nr = − kx a (k − 1) − kx where the summations are taken over k from 1 to n1 + · · · + nr . Proof. From Corollary 2 given in Mihoubi [7], the compositional inverse function of X nx tn F (t) = fn−1 (a (n − 1) + x) a (n − 1) + x n! n≥1

is given by F h−1i (t) = −

X

n≥1

tn nx fn−1 (a (n − 1) − nx) . a (n − 1) − nx n!

Then, it suffices to use the last function F in the pair of inverse relations given by (4). Remark 3. For x = 2a in (9), we obtain the remarkable inverse relations yn1 ,...,nr = xn1 ,...,nr =

n1 +···+n X r k=1 n1 +···+n X r k=1

2k fk−1 (a (k + 1)) Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , k+1 2k fk−1 (−a (k + 1)) Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) . k+1

We have binomial-type sequence of polynomials for fn (x) = xn , fn (x) = (αx)(n) := αx (αx − 1) · · · (αx − n + 1) for n ≥ 1 and (x)(0) := 1, fn (x) = (αx)(n) := αx (αx + 1) · · · (αx + n − 1) for n ≥ 1 and (x)(0) := 1,       n X 1 1 x Bn,j ( , . . . , i! , . . .)(x)(j) , fn (x) = n! := n q 1 q i q j=o n   X n k fn (x) = Bn (x) := x , k j=o n  where Bn (.), k and nk q are, respectively, the single variable Bell polynomials, the Stirling numbers of second kind and the ordinary multinomials.

191

Polynomials of multipartitional type and inverse relations

The ordinary multinomials, see Belbachir et al. [1], are defined by the following 2

q k

(1 + x + x + · · · + x ) =

X k  n≥0

n

xn , q

   with La 1 = La ( La being the usual binomial coefficient) and a > qL. Using the classical binomial coefficient, one has   L = a q

X

j1 +j2 +···+jq =a

L a q



= 0 for

     L j1 jq−1 ··· . j1 j2 jq

This last identity justify the extension of ordinary multinomials to real or complex values by replacing L by a real or complex value, one can see [2]. We deduce the following results: Corollary 6. Let a, α(6= 0) and x be real numbers, for fn (x) = xn , we get

yn1 ,...,nr =

n1 +···+n X r

xk (a (k − 1) + x)k−2 ×

k=1

(10)

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , xn1 ,...,nr = −

n1 +···+n X r

xk (a (k − 1) − kx)k−2 ×

k=1

Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) , for fn (x) = (x)(n) , we get

yn1 ,...,nr =

n1 +···+n X r k=1

kx (a (k − 1) + x)(k−1) × a (k − 1) + x

(11)

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , xn1 ,...,nr = −

n1 +···+n X r k=1

kx (a (k − 1) − kx)(k−1) × a (k − 1) − kx Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) ,

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M. Mihoubi and H. Belbachir

for fn (x) = (x)(n) , we get

yn1 ,...,nr =

n1 +···+n X r k=1

kx (a (k − 1) + x)(k−1) × a (k − 1) + x

(12)

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , xn1 ,...,nr = −

n1 +···+n X r k=1

kx (a (k − 1) − kx)(k−1) × a (k − 1) − kx Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) ,

  x for fn (x) = n! , we get n q

yn1 ,...,nr =

n1 +···+n X r k=1

  a (k − 1) + x k!x × a (k − 1) + x k−1 q

(13) xn1 ,...,nr = −

n1 +···+n X r k=1

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) ,   k!x a (k − 1) − kx × a (k − 1) − kx k−1 q Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) ,

and for fn (x) = Bn (x) , we get

yn1 ,...,nr =

n1 +···+n X r k=1

kx Bk−1 (a (k − 1) + x) × a (k − 1) + x

(14)

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , xn1 ,...,nr = −

n1 +···+n X r k=1

kx Bk−1 (a (k − 1) − kx) × a (k − 1) − kx Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) .

Example 1. For a = 0, x = 1 in (11) and using the fact that Bn1 ,...,nr ,1 (x0,...,0,1 , . . . , xn1 ,...,nr ) = xn1 ,...,nr and

Polynomials of multipartitional type and inverse relations

1 Bn1 ,...,nr ,2 (x0,...,0,1 , . . . , xn1 ,...,nr ) = 2



X

1≤i1 +···+ir ≤n1 +···+nr −1

193

   n1 nr ··· i1 ir

× xi1 ,...,ir xn1 −i1 ,...,nr −ir , we obtain yn1 ,...,nr = xn1 ,...,nr + X

1≤i1 +···+ir ≤n1 +···+nr −1

xn1 ,...,nr =

n1 +···+n X r

(−1)k−1

k=1

    n1 nr ··· xi1 ,...,ir xn1 −i1 ,...,nr −ir , i1 ir

(2k − 2)! Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) . (k − 1)!

Now, consider δ : N → {0, 1} be the application satisfying δ (0) = 0 and δ (n) = 1 for n ≥ 1. Theorem 7. Let u, r, s be nonnegative integers urs 6= 0, and {un } be a sequence of real numbers with u1 = 1. Then yn1 ,...,nr = s

n1 +···+n X r k=1

(15)

k BUk +k−1,Uk (1, u2 , u3 , . . .) × Uk +k−1 Uk Uk

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) ,

xn1 ,...,nr = yn1 ,...,nr − δ × s

n1 +···+n X r k=2

k BVk +k−1,Vk (1, u2 , u3 , . . .) ×  Vk +k−1 Vk V k

Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) ,

where δ = δ (n1 + · · · + nr − 1) , Uk = (r + 2s) (k − 1)+s and Vk = (r + s) (k − 1)− s.  1 (r+1)n −1 Proof. Let zn (r) := nr B(r+1)n,nr (1, u2 , u3 , . . .) , n ≥ 1, r integer ≥ 1, nr and let the binomial type polynomials {fn (x)} defined by fn (x) :=

n X k=1

Bn,k (z1 (r) , z2 (r) , . . .) xk , n ≥ 1, with f0 (x) = 1,

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see [10]. Then from [6, 8] or [7], we have the identity n X

Bn,k (z1 (r) , z2 (r) , . . .) sk

k=1

  s (r + 1) n + s −1 = B(r+1)n+s, nr + s nr + s

nr+s (1, u2 , u3 , . . .) ,

where r is an integer ≥ 1 and s is an integer, such that nr + s ≥ 1. Then we get   (r + 1) n + s −1 s B(r+1)n+s,nr+s (1, u2 , u3 , . . .) . (16) fn (s) = nr + s nr + s To obtain (15), it suffices to replace in (9) a by zero, fn (x) by the expression of fn (s) given by (16) and r by r + 2s.  Example 2. From the well-known identity Bn,k (1!, . . . , i!, . . .) = nk (n−1)! (k−1)! , we get yn1 ,...,nr = s

n1 +···+n X r

k

k=1

((r + 2s + 1) (k − 1) + s − 1)! × ((r + 2s) (k − 1) + s)! Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr )

xn1 ,...,nr = yn1 ,...,nr − δ × s

n1 +···+n X r k=2

k

((r + s + 1) (k − 1) − s − 1)! × ((r + s) (k − 1) − s)! Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) ,

where δ = δ (n1 + · · · + nr − 1) . We can obtain similar relations for the Stirling numbers (respectively the absolute Stirling numbers) of the first kind and the Stirling numbers of the second kind by setting un = (−1)n−1 (n − 1)! (respectively un = (n − 1)!) and un = 1, n ≥ 1. Corollary 8. Let u, r, s be nonnegative integers, a, α be real numbers, such that αurs 6= 0, and {fn (x)} be a polynomial sequence of binomial type. Then

yn1 ,...,nr = s

n1 +···+n X r k=1

Tk kDz=0 (eαz fk−1 (Tk x + z; a)) × αTk −u(k−1) Tk

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , xn1 ,...,nr = yn1 ,...,nr − δs

n1 +···+n X r k=2

Rk kDz=0 (eαz fk−1 (Rk x + z; a)) × αRk −u(k−1) Rk

Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) ,

Polynomials of multipartitional type and inverse relations

195

where δ = δ (n1 + · · · + nr − 1) , Tk = (u + r + 2s) (k − 1)+s and Rk = (u + r + s) (k − 1)− s. Proof. In Theorem 7 set un =

n u(n−1)+1 αz D (e fn−1 ((u (n − 1) + 1) x + z; a)) (u (n − 1) + 1) α z=0

and use the first identity of [8, Theorem 2]. Corollary 9. Let u, r, s be nonnegative integers, such that urs 6= 0, a be a real number and {fn (x)} be a binomial type polynomials. Then yn1 ,...,nr = s

n1 +···+n X r k=1

Tk k!Dz=0 fTk +k−1 (Tk x + z; a) × T −u(k−1) α k (Tk + k − 1)!Tk

Bn1 ,...,nr ,k (x0,...,0,1 , . . . , xn1 ,...,nr ) , xn1 ,...,nr = yn1 ,...,nr − δ × s

n1 +···+n X r k=2

Rk k!Dz=0 fRk +k−1 (Rk x + z; a) × R −u(k−1) α k (Rk + k − 1)!Rk

Bn1 ,...,nr ,k (y0,...,0,1 , . . . , yn1 ,...,nr ) , where δ = δ (n1 + · · · + nr − 1) , Tk = (u + r + 2s) (k − 1)+s and Rk = (u + r + s) (k − 1)− s. Proof. In Theorem 7 set u(n−1)+1

un =

f((u+1)(n−1)+1) ((u (n − 1) + 1) x + z; a) n!Dz=0 ((u + 1) (n − 1) + 1)! (u (n − 1) + 1) α

and use the second identity of [8, Theorem 2].

3.

Polynomials of multipartitional type and inverse relations

For any sequence u = (un1 ,...,nr ) of real numbers, let ϕu (t) := ϕu (t1 , . . . , tr ) =

X

n1 +···+nr ≥1

un1 ,...,nr

tn1 1 tnr ··· r n1 ! nr !

    (1) (s) and for any sequences u(1) = un1 ,...,nr , . . . , u(s) = un1 ,...,nr of real numbers, let be defined the polynomials of multipartitional type   Bn1 ,...,nr / j1 ,...,js u(1) , . . . , u(s)

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by X

(17)

Bn1 ,...,nr

n1 +···+nr ≥j1 +···+js ≥1

:=

/ j1 ,...,js



u(1) , . . . , u(s)

 tn1 tnr 1 ··· r n1 ! nr !

(ϕu(1) (t1 , . . . , tr ))j1 (ϕ (s) (t1 , . . . , tr ))js ··· u , j1 ! js !

with Bn1 ,...,nr

/

   1 if n + · · · + n = 0, 1 r (1) (s) = 0,...,0 u , . . . , u 0 if n1 + · · · + nr ≥ 1.

Theorem 10. Let a = (a1 , . . . , as ) be a vector of Rs and F = (F1 , . . . , Fs ) : I → J be a bijective function with I, J ⊆ Rs such that F (a + x) admits formal power series in each composite xk of x and let G = (G1 , . . . , Gs ) denote the compositional inverse of F so that (F ◦ G) (x) = (G ◦ F ) (x) = x. Then, we have the vectorial pair of inverse relations

u(1) n1 ,...,nr =

X ∂F j1 +···+js 1

(a) Bn1 ,...,nr

X ∂Fsj1 +···+js

(a) Bn1 ,...,nr

J

∂xj11 · · · ∂xjss

/

  (1) (s) v , . . . , v , j1 ,...,js

/

  (1) (s) v , . . . , v , j1 ,...,js

.. . u(s) n1 ,...,nr =

J

∂xj11 · · · ∂xjss

if and only if

vn(1) = 1 ,...,nr

X ∂Gj1 +···+js 1

(F (a)) Bn1 ,...,nr

/ j1 ,...,js

  u(1) , . . . , u(s) ,

X ∂Gjs1 +···+js

(F (a)) Bn1 ,...,nr

/ j1 ,...,js

  u(1) , . . . , u(s) ,

J

∂xj11 · · · ∂xjss

.. . vn(s) = 1 ,...,nr

J

∂xj11 · · · ∂xjss

where J = {(j1 , . . . , js ) ∈ Ns : 1 ≤ j1 + · · · + js ≤ n1 + · · · + nr }.

Polynomials of multipartitional type and inverse relations

Proof. We have

ϕu(k) (t) =

X

un1 ,...,nr

n1 +···+nr ≥1

=

X

n1 +···+nr ≥1

tn1 1 tnr ··· r n1 ! nr !

tn1 1 tnr ··· r n1 ! nr !

j1 j1 +···+jk ≤n1 +···+nr ∂x1

× Bn1 ,...,nr

=

X

· · · ∂xjss

× Bn1 ,...,nr

=

X

/ j1 ,...,js

∂Fkj1 +···+js

j1 j1 +···+jk ≥1 ∂x1

(a)

· · · ∂xjss

· · · ∂xjss

(a)

  v (1) , . . . , v (s) X

n1 +···+nr ≥j1 +···+jk

/ j1 ,...,js

∂Fkj1 +···+js

j1 j1 +···+jk ≥1 ∂x1

∂Fkj1 +···+js

X

tn1 1 tnr ··· r n1 ! nr !

  v (1) , . . . , v (s)

(ϕv(1) (t))j1 (ϕv(s) (t))js (a) ··· j1 ! js !

= Fk (a+ (ϕv(1) (t) , . . . , ϕv(s) (t))) − Fk (a)

which implies

(F1 (a) + ϕu(1) (t) , . . . , Fs (a) + ϕu(s) (t)) = (F1 (a+ (ϕv(1) (t) , . . . , ϕv(s) (t))) , . . . , Fs (a+ (ϕv(1) (t) , . . . , ϕv(s) (t))))

or equivalently

F (a) + (ϕu(1) (t) , . . . , ϕu(s) (t)) = F (a1 + ϕv(1) (t) , . . . , as + ϕv(s) (t)) .

197

198

M. Mihoubi and H. Belbachir

Then, applying G to the two sides of this equality, we obtain ϕv(k) (t) = =

∂Gkj1 +···+js

X

j1 js j1 +···+js ≥1 ∂x1 · · · ∂xs X ∂Gkj1 +···+js j1 js j1 +···+js ≥1 ∂x1 · · · ∂xs

X

=

n1 +···+nr ≥1

tn1 1 tnr ··· r n1 ! nr !

(ϕu(1) (t))j1 (ϕu(s) (t))js (F (a)) ··· j1 ! js ! tnr tn1 1 ··· r × n ! nr ! n1 +···+nr ≥j1 +···+js 1   Bn1 ,...,nr / j1 ,...,js u(1) , . . . , u(s) X

(F (a))

∂Gjk1 +···+js

X

(F (a)) × j1 js ∂x · · · ∂x s 1 1≤j1 +···+js ≤n1 +···+nr   Bn1 ,...,nr / j1 ,...,js u(1) , . . . , u(s) .

Corollary 11. Let A = (αij )1≤i,j≤s be a nonsingular matrix of real numbers and B = A−1 = (βij ) be the inverse matrix of A. Then we have the vectorial pair of inverse relations  P (1) un1 ,...,nr = (α11 )j1 · · · (α1s )js Bn1 ,...,nr / j1 ,...,js v (1) , . . . , v (s) , J

(s)

un1 ,...,nr

.. . P = (αs1 )j1 · · · (αss )js Bn1 ,...,nr J

/ j1 ,...,js

 v (1) , . . . , v (s) ,

if and only if vn(1) = 1 ,...,nr

  (1) (s) u , . . . , u , j1 ,...,js

X

(β11 )j1 · · · (β1s )js Bn1 ,...,nr

/

X

(βs1 )j1 · · · (βss )js Bn1 ,...,nr

/ j1 ,...,js

J

.. . vn(s) = 1 ,...,nr

J

  u(1) , . . . , u(s) .

Proof. The corollary follows by setting, in Theorem 10, a = (1, . . . , 1) and F (x) = (xα1 11 · · · xαs 1s , . . . , xα1 s1 · · · xαs ss ) . Remark 4. If we set s = 1 in Theorem 10, we obtain Theorem 1, and, if we set s = r = 1 in Theorem 10, we obtain Theorem 1 given in [7].

References

Polynomials of multipartitional type and inverse relations

199

[1] H. Belbachir, S. Bouroubi and A. Khelladi, Connection between ordinary multinomials, generalized Fibonacci numbers, partial Bell partition polynomials and convolution powers of discrete uniform distribution, Ann. Math. Inform. 35 (2008), 21–30. [2] H. Belbachir, Determining the mode for convolution powers of discrete uniform distribution, Probability in the Engineering and Informational Sciences 25 (2011), 469– 475. doi:10.1017/S0269964811000131 [3] E.T. Bell, Exponential polynomials, Ann. Math. 35 (1934), 258–277. doi:10.2307/ 1968431 [4] W.S. Chou, L.C. Hsu and P.J.S. Shiue, Application of Fa` a di Bruno’s formula in characterization of inverse relations, J. Comput. Appl. Math. 190 (2006), 151–169. doi:10.1016/j.cam.2004.12.041 [5] L. Comtet, Advanced Combinatorics (Dordrecht, Netherlands, Reidel, 1974). doi:10. 1007/978-94-010-2196-8 [6] M. Mihoubi, Bell polynomials and binomial type sequences, Discrete Math. 308 (2008), 2450–2459. doi:10.1016/j.disc.2007.05.010 [7] M. Mihoubi, Bell polynomials and inverse relations, J. Integer Seq. 13 (2010), Article 10.4.5. [8] M. Mihoubi, The role of binomial type sequences in determination identities for Bell polynomials, to appear in Ars Combin., Preprint available at online: http://arxiv.org/abs/0806.3468v1. [9] J. Riordan, Combinatorial Identities (Huntington, NewYork, 1979). [10] S. Roman, The Umbral Calculus (New York: Academic Press, 1984). Received 24 February 2011 Revised 6 September 2011