The Compositions of Differential Operations and the Gateaux Directional Derivative

Branko J. Malešević^{1} and Ivana V. Jovović^{2} University of Belgrade Faculty of Electrical Engineering Bulevar kralja Aleksandra 73 Belgrade Serbia

                        malesh@EUnet.yu
                       ivana121@EUnet.yu

Abstract This paper deals with the number of meaningful compositions of higher order of

differential operations and the Gateaux directional derivative.

1 The compositions of differential operations of the space ℝ^{3}

In the real three-dimensional space ℝ^{3} we consider the following sets:

A_{0} = {f:ℝ^{3} ⟶ℝ | f ∈C^{∞}(ℝ^{3})} and A_{1} = {f⃗:ℝ^{3} ⟶ℝ^{3} | f⃗∈C⃗^{∞}(ℝ^{3})}.  (1)

It is customary in vector analysis to consider m = 3 basic differential operations on A_{0} and A_{1} [1], namely:

grad f = ∇_{1} f =(\frac{∂f}{∂x_{1}} , \frac{∂f}{∂x_{2}} , \frac{∂f}{∂x_{3}} ) : A_{0} ⟶ A_{1},

curl f⃗ = ∇_{2} f⃗ = (\frac{∂f_{3}}{∂x_{2}} − \frac{∂f_{2}}{∂x_{3}} , \frac{∂f_{1}}{∂x_{3}} − \frac{∂f_{3}}{∂x_{1}} , \frac{∂f_{2}}{∂x_{1}} − \frac{∂f_{1}}{∂x_{2}} ) : A_{1} ⟶ A_{1},

div f⃗ = ∇_{3} f⃗ = \frac{∂f_{1}}{∂x_{1}} + \frac{∂f_{2}}{∂x_{2}} + \frac{∂f_{3}}{∂x_{3}} : A_{1} ⟶ A_{0} .

(2)

1Supported in part by the project MNTRS, Grant No. ON144020.

2Ph. D. student, Faculty of Mathematics, University of Belgrade, Serbia.

Let us present the number of meaningful compositions of higher order over the set A_{3} = {∇_{1}, ∇_{2}, ∇_{3}}. It is familiar fact that there are m = 5 compositions of the second order [2, p. 161]:

Δf = div grad f = ∇_{3} ∘ ∇_{1} f,
curl curl f⃗ = ∇_{2} ∘ ∇_{2} f⃗,
grad div f⃗ = ∇_{1} ∘ ∇_{3} f⃗,
curl grad f = ∇_{2} ∘ ∇_{1} f = 0,
div curl f⃗ = ∇_{3} ∘ ∇_{2} f⃗ = 0.

(3)

Malešević [3] proved that there are m = 8 compositions of the third order:

grad div grad f = ∇_{1} ∘ ∇_{3} ∘ ∇_{1} f,
curl curl curl f⃗ = ∇_{2} ∘ ∇_{2} ∘ ∇_{2} f⃗,
div grad div f⃗ = ∇_{3} ∘ ∇_{1} ∘ ∇_{3} f⃗,
curl curl grad f = ∇_{2} ∘ ∇_{2} ∘ ∇_{1} f = 0,
div curl grad f = ∇_{3} ∘ ∇_{2} ∘ ∇_{1} f = 0,
div curl curl f⃗ = ∇_{3} ∘ ∇_{2} ∘ ∇_{2} f⃗ = 0,
grad div curl f⃗ = ∇_{1} ∘ ∇_{3} ∘ ∇_{2} f⃗ = 0,
curl grad div f⃗ = ∇_{2} ∘ ∇_{1} ∘ ∇_{3} f⃗ = 0.

(4)

If f(k) is the number of compositions of the k^{th} order, then Malešević [4] proved

f(k) = F_{k+3},  (5)

where F_{k} is k^{th} Fibonacci number.

2 The compositions of the differential operations and Gateaux directional derivative of the space ℝ^{3}

Let f ∈ A_{0} be a scalar function and e⃗ = (e_{1}, e_{2}, e_{3}) ∈ ℝ^{3} be a unit vector. The Gateaux directional derivative in direction e⃗ is defined by [5, p. 71]:

dir_{e⃗} f = ∇_{0}f = ∇_{1}f · e⃗ = \frac{∂f}{∂x_{1}} e_{1} + \frac{∂f}{∂x_{2}} e_{2} + \frac{∂f}{∂x_{3}} e_{3} : A_{0} ⟶ A_{0}.  (6)

Let us determine the number of meaningful compositions of higher order over the set B_{3} = {∇_{0}, ∇_{1}, ∇_{2}, ∇_{3}}. There exist m = 8 compositions of the second order:

dir_{e⃗} dir_{e⃗} f = ∇_{0} ∘ ∇_{0} f = ∇_{1}( ∇_{1}f · e⃗ ) · e⃗,

grad dir_{e⃗} f = ∇_{1} ∘ ∇_{0} f = ∇_{1}( ∇_{1}f · e⃗ ),

Δf = div grad f = ∇_{3} ∘ ∇_{1} f,

curl curl f⃗ = ∇_{2} ∘ ∇_{2} f⃗,

dir_{e⃗} div f⃗ = ∇_{0} ∘ ∇_{3} f⃗ = (∇_{1} ∘ ∇_{3} f⃗ ) · e⃗,

grad div f⃗ = ∇_{1} ∘ ∇_{3} f⃗,

curl grad f = ∇_{2} ∘ ∇_{1} f = 0,

div curl f⃗ = ∇_{3} ∘ ∇_{2} f⃗ = 0;

(7)

and there exist m = 16 compositions of the third order:

dir_{e⃗} dir_{e⃗} dir_{e⃗} f = ∇_{0} ∘ ∇_{0} ∘ ∇_{0} f,

grad dir_{e⃗} dir_{e⃗} f = ∇_{1} ∘ ∇_{0} ∘ ∇_{0} f,

div grad dir_{e⃗} f = ∇_{3} ∘ ∇_{1} ∘ ∇_{0} f,

dir_{e⃗} div grad f = ∇_{0} ∘ ∇_{3} ∘ ∇_{1} f,

grad div grad f = ∇_{1} ∘ ∇_{3} ∘ ∇_{1} f,

curl curl curl f⃗ = ∇_{2} ∘ ∇_{2} ∘ ∇_{2} f⃗,

dir_{e⃗} dir_{e⃗} div f⃗ = ∇_{0} ∘ ∇_{0} ∘ ∇_{3} f⃗,

grad dir_{e⃗} div f⃗ = ∇_{1} ∘ ∇_{0} ∘ ∇_{3} f⃗,

div grad div f⃗ = ∇_{3} ∘ ∇_{1} ∘ ∇_{3} f⃗,

curl grad dir_{e⃗} f = ∇_{2} ∘ ∇_{1} ∘ ∇_{0} f⃗ = 0,

curl curl grad f = ∇_{2} ∘ ∇_{2} ∘ ∇_{1} f = 0,

div curl grad f = ∇_{3} ∘ ∇_{2} ∘ ∇_{1} f = 0,

div curl curl f⃗ = ∇_{3} ∘ ∇_{2} ∘ ∇_{2} f⃗ = 0,

dir_{e⃗} div curl f⃗ = ∇_{0} ∘ ∇_{3} ∘ ∇_{2} f⃗ = 0,

grad div curl f⃗ = ∇_{1} ∘ ∇_{3} ∘ ∇_{2} f⃗ = 0,

curl grad div f⃗ = ∇_{2} ∘ ∇_{1} ∘ ∇_{3} f⃗ = 0.

(8)

Further on we shall use the method from the paper [4]. Let us define a binary relation σ “to be in composition”: ∇_{i} σ ∇_{j} iff the composition ∇_{j} ∘ ∇_{i} is meaningful. Then Cayley table of the relation σ is determined by

\begin{matrix}σ & ∇_{0} & ∇_{1} & ∇_{2} & ∇_{3} \\ ∇_{0} & ⊤ & ⊤ & ⊥ & ⊥ \\ ∇_{1} & ⊥ & ⊥ & ⊤ & ⊤ \\ ∇_{2} & ⊥ & ⊥ & ⊤ & ⊤ \\ ∇_{3} & ⊤ & ⊤ & ⊥ & ⊥ \end{matrix}  (9)

Let us denote by ∇_{−1} nowhere-defined function, where domain and range are empty sets [3] and let ∇_{−1} σ ∇_{i} hold for i = 0, 1, 2, 3. If G is graph which is determined by the relation σ, then graph of paths of G is the tree with the root ∇_{−1} (Fig. 1).

[figure: ∇_{−1}; g(0) = 1; ∇_{0}; ∇_{1}; ∇_{2}; ∇_{3}; g(1) = 4; ∇_{0}; ∇_{1}; ∇_{2}; ∇_{3}; ∇_{2}; ∇_{3}; ∇_{0}; ∇_{1}; g(2) = 8; g(3) = 16; Fig. 1]

Let g(k) be the number of meaningful compositions of the k^{th} order of the functions from B_{3} and let g_{i}(k) be the number of meaningful compositions of the k^{th} order beginning from the left by ∇_{i}. Then g(k) = g_{0}(k) + g_{1}(k) + g_{2}(k) + g_{3}(k). Based on the partial self similarity of the tree (Fig. 1) we obtain equalities

g_{0}(k) = g_{0}(k − 1) + g_{1}(k − 1),
g_{1}(k) = g_{2}(k − 1) + g_{3}(k − 1),
g_{2}(k) = g_{2}(k − 1) + g_{3}(k − 1),
g_{3}(k) = g_{0}(k − 1) + g_{1}(k − 1).

(10)

Hence, the recurrence for g(k) is

g(k)=2 g(k − 1)  (11)

and because g(1) = 4 we have

g(k) = 2^{k+1}.  (12)

3 The compositions of differential operations of the space ℝ^{n}

Let us present the number of meaningful compositions of differential operations in the vector analysis of the space ℝ^{n}, where differential operations ∇_{r} (r = 1, …, n) are defined on corresponding non-empty sets A_{s} (s = 1, …, m and m = ⌊n/2⌋, n ≥ 3) according to the papers [4], [6]:

A_{n} (n=2m): ∇_{1} : A_{0} →A_{1} ∇_{2} : A_{1} →A_{2} ⋮ ∇_{i} : A_{i−1} →A_{i} ⋮ ∇_{m} : A_{m−1} →A_{m} ∇_{m+1} : A_{m} →A_{m−1} ⋮ ∇_{n−j} : A_{j+1} →A_{j} ⋮ ∇_{n−1} : A_{2} →A_{1} ∇_{n} : A_{1} →A_{0}, A_{n} (n=2m+1): ∇_{1} : A_{0} →A_{1} ∇_{2} : A_{1} →A_{2} ⋮ ∇_{i} : A_{i−1} →A_{i} ⋮ ∇_{m} : A_{m−1} →A_{m} ∇_{m+1} : A_{m} →A_{m} ∇_{m+2} : A_{m} →A_{m−1} ⋮ ∇_{n−j} : A_{j+1} →A_{j} ⋮ ∇_{n−1} : A_{2} →A_{1} ∇_{n} : A_{1} →A_{0}.  (13)

Let us define higher order differential operations as meaningful compositions of higher order of differential operations from the set A_{n} = {∇_{1}, …, ∇_{n}}. The number of higher order differential operations is given according to the paper [4]. Furthermore, let us define a binary relation ρ “to be in composition”: ∇_{i} ρ ∇_{j} iff the composition ∇_{j} ∘ ∇_{i} is meaningful. Then Cayley table of the relation ρ is determined by

∇_{i} ρ ∇_{j} = \{ \begin{matrix}⊤ & , & (j = i + 1) ∨ (i + j = n + 1); \\ ⊥ & , & otherwise. \end{matrix}  (14)

Let A = [a_{ij}] ∈ { 0, 1}^{n×n} be the adjacency matrix associated with the graph which is determined by the relation ρ. Malešević [6] proved the following statements.

Theorem 3.1. Let P_{n}(λ)=|A−λI|=α_{0}λ^{n}+α_{1}λ^{n−1}+⋯+α_{n} be the characteristic polynomial of the matrix A and v_{n} = [ 1 … 1 ]_{1×n}. If f(k) is the number of the k^{th} order differential operations, then the following formulas hold:

f(k) = v_{n} · A^{k−1} · v_{n}^{T}  (15)

and

α_{0}f(k) + α_{1}f(k − 1) + ⋯ + α_{n}f(k − n) = 0 (k > n).  (16)

Lemma 3.2. Let P_{n}(λ) be the characteristic polynomial of the matrix A. Then the following recurrence holds:

P_{n}(λ) = λ^{2}(P_{n−2}(λ) − P_{n−4}(λ)).  (17)

Lemma 3.3. Let P_{n}(λ) be the characteristic polynomial of the matrix A. Then it has the following explicit form:

P_{n}(λ) = \begin{cases}∑_{k=1}^{⌊ \frac{n+2}{4} ⌋+1} (−1)^{k−1}( \begin{matrix}\frac{n}{2} −k+2 \\ k−1 \end{matrix} )λ^{n−2k+2} & , & n=2m; \\ ∑_{k=1}^{⌊ \frac{n+2}{4} ⌋+2}(−1)^{k−1}(( \begin{matrix}\frac{n+3}{2} −k \\ k−1 \end{matrix} )+( \begin{matrix}\frac{n+3}{2} −k \\ k−2 \end{matrix} )λ)λ^{n−2k+2} & , & n=2m+1. \end{cases} (18)

From previous statements one can obtain the recurrences in the table, [4]:

Dimension  Recurrence for the number of the k^{th} order differential operations
n = 3  f(k) = f(k − 1) + f(k − 2)
n = 4  f(k) = 2f(k − 2)
n = 5  f(k) = f(k − 1) + 2f(k − 2) − f(k − 3)
n = 6  f(k) = 3f(k − 2) − f(k − 4)
n = 7  f(k) = f(k − 1) + 3f(k − 2) − 2f(k − 3) − f(k − 4)
n = 8  f(k) = 4f(k − 2) − 3f(k − 4)
n = 9  f(k) = f(k − 1) + 4f(k − 2) − 3f(k − 3) − 3f(k − 4) + f(k − 5)
n = 10  f(k) = 5f(k − 2) − 6f(k − 4) + f(k − 6)

The values of the function f(k), for small values of the argument k, are given in the database of integer sequences [8] as the following sequences \underline{A020701} (n = 3), \underline{A090989} (n = 4), \underline{A090990} (n = 5), \underline{A090991} (n = 6), \underline{A090992} (n = 7), \underline{A090993} (n = 8), \underline{A090994} (n = 9), \underline{A090995} (n = 10).

4 The compositions of differential operations and Gateaux directional derivative of the space ℝ^{n}

Let f ∈ A_{0} be a scalar function and e⃗ = (e_{1}, …, e_{n}) ∈ ℝ^{n} be a unit vector. The Gateaux directional derivative in direction e⃗ is defined by [5, p. 71]:

dir_{e⃗} f = ∇_{0}f = ∑_{k=1}^{n} \frac{∂f}{∂x_{k}} e_{k} : A_{0} ⟶ A_{0}.  (19)

Let us extend the set of differential operations A_{n} = {∇_{1}, …, ∇_{n}} with Gateaux directional derivative to the set B_{n} = A_{n} ∪ {∇_{0}} = {∇_{0}, ∇_{1}, …, ∇_{n}}:

B_{n} (n=2m): ∇_{0} : A_{0} →A_{0} ∇_{1} : A_{0} →A_{1} ∇_{2} : A_{1} →A_{2} ⋮ ∇_{i} : A_{i−1} →A_{i} ⋮ ∇_{m} : A_{m−1} →A_{m} ∇_{m+1} : A_{m} →A_{m−1} ⋮ ∇_{n−j} : A_{j+1} →A_{j} ⋮ ∇_{n−1} : A_{2} →A_{1} ∇_{n} : A_{1} →A_{0}, B_{n} (n=2m+1): ∇_{0} : A_{0} →A_{0} ∇_{1} : A_{0} →A_{1} ∇_{2} : A_{1} →A_{2} ⋮ ∇_{i} : A_{i−1} →A_{i} ⋮ ∇_{m} : A_{m−1} →A_{m} ∇_{m+1} : A_{m} →A_{m} ∇_{m+2} : A_{m} →A_{m−1} ⋮ ∇_{n−j} : A_{j+1} →A_{j} ⋮ ∇_{n−1} : A_{2} →A_{1} ∇_{n} : A_{1} →A_{0}.  (20)

Let us define higher order differential operations with Gateaux derivative as the meaningful compositions of higher order of the functions from the set B_{n} = {∇_{0}, ∇_{1}, …, ∇_{n}}. Our aim is to determine the number of higher order differential operations with Gateaux derivative. Let us define a binary relation σ “to be in composition”:

∇_{i} σ ∇_{j} = \{ \begin{matrix}⊤ & , & (i=0 ∧ j =0) ∨ (i=n ∧ j =0) ∨ (j =i+1) ∨ (i+j =n+1); \\ ⊥ & , & otherwise. \end{matrix}  (21)

and let B = [b_{ij}] ∈ { 0, 1}^{(n+1)×n} be the adjacency matrix associated with the graph which is determined by relation σ. So, analogously to the paper [6], the following statements hold.

Theorem 4.1. Let Q_{n}(λ) = |B−λI| = β_{0}λ^{n+1} + β_{1}λ^{n} + ⋯ + β_{n+1} be the characteristic polynomial of the matrix B and v_{n+1} = [ 1 … 1 ]_{1×(n+1)}. If g(k) is the number of the k^{th} order differential operations with Gateaux derivative, then the following formulas hold:

g(k) = v_{n+1} · B^{k−1} · v_{n+1}^{T}  (22)

and

β_{0}g(k) + β_{1}g(k − 1) + ⋯ + β_{n+1}g(k − (n + 1)) = 0 (k > n+1).  (23)

Lemma 4.2. Let Q_{n}(λ) and P_{n}(λ) be the characteristic polynomials of the matrices B and A respectively. Then the following equality holds:

Q_{n}(λ) = λ^{2}P_{n−2}(λ) − λP_{n}(λ).  (24)

Proof. Let us calculate the characteristic polynomial

Q_{n}(λ) = |B − λI| = | \begin{matrix}1 − λ & 1 & 0 & 0 & … & 0 & 0 & 0 & 0 \\ 0 & −λ & 1 & 0 & … & 0 & 0 & 0 & 1 \\ 0 & 0 & −λ & 1 & … & 0 & 0 & 1 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 1 & … & 0 & −λ & 1 & 0 \\ 0 & 0 & 1 & 0 & … & 0 & 0 & −λ & 1 \\ 1 & 1 & 0 & 0 & … & 0 & 0 & 0 & −λ \end{matrix} | .  (25)

Expanding the determinant Q_{n}(λ) by the first column we have

Q_{n}(λ) = (1 − λ)P_{n}(λ) + (−1)^{n+2}D_{n}(λ),  (26)

where

D_{n}(λ) = | \begin{matrix}1 & 0 & 0 & 0 & … & 0 & 0 & 0 & 0 \\ −λ & 1 & 0 & 0 & … & 0 & 0 & 0 & 1 \\ 0 & −λ & 1 & 0 & … & 0 & 0 & 1 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 1 & … & −λ & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & … & 0 & −λ & 1 & 0 \\ 0 & 1 & 0 & 0 & … & 0 & 0 & −λ & 1 \end{matrix} | .  (27)

Let us expand the determinant D_{n}(λ) by the first row and then in the next step, multiply the first row by −1 and add it to the last row. We obtain the determinant of order n − 1 :

D_{n}(λ) = | \begin{matrix}1 & 0 & 0 & 0 & … & 0 & 0 & 0 & 1 \\ −λ & 1 & 0 & 0 & … & 0 & 0 & 1 & 0 \\ 0 & −λ & 1 & 0 & … & 0 & 1 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 1 & 0 & … & −λ & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & … & 0 & −λ & 1 & 0 \\ 0 & 0 & 0 & 0 & … & 0 & 0 & −λ & 0 \end{matrix} | .  (28)

Expanding the previous determinant by the last column we have

D_{n}(λ) = (−1)^{n}| \begin{matrix}−λ & 1 & 0 & 0 & … & 0 & 0 & 0 & 1 \\ 0 & −λ & 1 & 0 & … & 0 & 0 & 1 & 0 \\ 0 & 0 & −λ & 1 & … & 0 & 1 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 1 & 0 & … & 0 & −λ & 1 & 0 \\ 0 & 1 & 0 & 0 & … & 0 & 0 & −λ & 1 \\ 0 & 0 & 0 & 0 & … & 0 & 0 & 0 & −λ \end{matrix} | .  (29)

If we expand the previous determinant by the last row and if we expand the obtained determinant by the first column, we have the determinant of order n − 4 :

D_{n}(λ) = (−1)^{n}λ^{2}| \begin{matrix}−λ & 1 & 0 & 0 & … & 0 & 0 & 0 & 1 \\ 0 & −λ & 1 & 0 & … & 0 & 0 & 1 & 0 \\ 0 & 0 & −λ & 1 & … & 0 & 1 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 1 & 0 & … & 0 & −λ & 1 & 0 \\ 0 & 1 & 0 & 0 & … & 0 & 0 & −λ & 1 \\ 1 & 0 & 0 & 0 & … & 0 & 0 & 0 & −λ \end{matrix} | .  (30)

In other words

D_{n}(λ) = (−1)^{n}λ^{2}P_{n−4}(λ).  (31)

From equalities (31) and (26) there follows:

Q_{n}(λ) = (1 − λ)P_{n}(λ) + λ^{2}P_{n−4}(λ).  (32)

On the basis of Lemma 3.2. the following equality holds:

Q_{n}(λ) = λ^{2}P_{n−2}(λ) − λP_{n}(λ).  (33)

Lemma 4.3. Let Q_{n}(λ) be the characteristic polynomial of the matrix B. Then the following recurrence holds:

Q_{n}(λ) = λ^{2}(Q_{n−2}(λ) − Q_{n−4}(λ)).  (34)

Proof. On the basis of Lemma 3.2. and Lemma 4.2. the Lemma follows.

Lemma 4.4. Let Q_{n}(λ) be the characteristic polynomial of the matrix B. Then it has the following explicit form:

Q_{n}(λ) = \begin{cases}(λ − 2) ∑_{k=1}^{⌊ \frac{n}{4}⌋+1} (−1)^{k−1}( \begin{matrix}\frac{n + 1}{2} −k \\ k−1 \end{matrix} )λ^{n−2k+2} & , & n=2m+1; \\ ∑_{k=1}^{⌊ \frac{n+3}{4} ⌋+2}(−1)^{k−1}(\binom{\frac{n}{2} −k+2}{k−1}+( \begin{matrix}\frac{n}{2} −k+2 \\ k−2 \end{matrix} )λ)λ^{n−2k+3} & , & n=2m. \end{cases}  (35)

Proof. On the basis of Lemma 3.3 and Lemma 4.2. the Lemma follows.

The recurrences for dimensions n = 3, 4, …, 10 are obtained by means of Malešević-Jovović [7] and they are given in the table below.

Dimension Recurrence for the num. of the k^{th} order diff. operations with Gateaux derivative
n = 3  g(k) = 2g(k − 1)
n = 4  g(k) = g(k − 1) + 2g(k − 2) − g(k − 3)
n = 5  g(k) = 2g(k − 1) + g(k − 2) − 2g(k − 3)
n = 6  g(k) = g(k − 1) + 3g(k − 2) − 2g(k − 3) − g(k − 4)
n = 7  g(k) = 2g(k − 1) + 2g(k − 2) − 4g(k − 3)
n = 8  g(k) = g(k − 1) + 4g(k − 2) − 3g(k − 3) − 3g(k − 4) + g(k − 5)
n = 9  g(k) = 2g(k − 1) + 3g(k − 2) − 6g(k − 3) − g(k − 4) + 2g(k − 5)
n =10  g(k) = g(k − 1) + 5g(k − 2) − 4g(k − 3) − 6g(k − 4) + 3g(k − 5) + g(k − 6)

The values of the function g(k), for small values of the argument k, are given in the database of integer sequences [8] as the following sequences \underline{A000079} (n = 3), \underline{A090990} (n = 4), \underline{A007283} (n = 5), \underline{A090992} (n = 6), \underline{A000079} (n = 7), \underline{A090994} (n = 8), \underline{A020714} (n = 9), \underline{A129638} (n = 10).

References

[1] Ivanov, A. B., Vector analysis, in M. Hazewinkel, ed., Encyclopaedia of Mathematics,

Springer, 2002. Text available at http://eom.springer.de/V/v096360.htm.

[2] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Courier Dover Publications, 2000.

[3] B. J. Malešević, A note on higher-order differential operations, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 7 (1996), 105–109. Text available at

http://pefmath2.etf.bg.ac.yu/files/116/846.pdf.

[4] B. J. Malešević, Some combinatorial aspects of differential operation composition on the space R^{n}, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29–33. Text

available at http://pefmath2.etf.bg.ac.yu/files/118/869.pdf.

[5] S. Basov, Multidimensional Screening, Springer 2005.

[6] B. J. Malešević, Some combinatorial aspects of the composition of a set of function, Novi Sad J. Math., 36 (1), 2006, 3–9. Text available at

 http://www.im.ns.ac.yu/NSJOM/Papers/36 1/NSJOM 36 1 003 009.pdf.

[7] B. J. Malešević and I. V. Jovović, A procedure for finding the k^{th} power of a matrix.

http://www.maplesoft.com/Applications.

[8] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.

http://www.research.att.com/~njas/sequences/.

2000 Mathematics Subject Classification: 05C30, 26B12, 58C20.

Keywords: compositions of the differential operations, enumeration of graphs and maps, Gateaux directional derivative.

(Concerned with sequences \underline{A000079}, \underline{A007283}, \underline{A020701}, \underline{A020714}, \underline{A090989}, \underline{A090990}, \underline{A090991}, \underline{A090992}, \underline{A090993}, \underline{A090994}, \underline{A090995}, and \underline{A129638}.)

Received June 5, 2007; revised version received July 30, 2007. Published in Journal of Integer Sequences, August 3 2007.

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