A study on partial dynamic equation on time scales involving derivatives of polynomials

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A study on partial dynamic equation on time scales involving derivatives of polynomials

Petro Kolosov

To cite this version:

Petro Kolosov. A study on partial dynamic equation on time scales involving derivatives of polyno- mials. 2020. �hal-02556096�

A STUDY ON PARTIAL DYNAMIC EQUATION ON TIME SCALES INVOLVING DERIVATIVES OF POLYNOMIALS

PETRO KOLOSOV

Abstract. Let fm(x, b) be a 2m+ 1−degree integer-valued polynomial inx, b. Let be a two-dimensional time scale Λ²=T₁×T₂:={t= (x, b) : x∈T₁, b∈T₂}. Let be T₁=T₂. For everyx, b∈Λ², m∈N

∂fm(x, b)

∆x (x0, σ(x0)) +∂fm(x, b)

∆b (x0, x0) = (x^2m+1)^∆(x0),

whereσ(x)> xis forward jump operator. In this manuscript we derive and discuss above partial differential equation on time scales and show few polynomial identities. In addition, we discus various derivative operators in context of partial cases of above equation, we show finite difference, classical derivative,q−derivative,q−power derivative on behalf of it.

Contents

1. Definitions, Notations and Conventions 1

2. Introduction 3

3. Main results 3

4. Discussion and examples 5

4.1. Time scale T=Z×Z 5

4.2. Time scale T=R×R 6

4.3. Time scale T=q^R×q^R 7

4.4. Time scale T=R^q×R^q 8

4.5. Time scale T=q^Rn ×q^Rn 10

5. Proof of main theorem 11

Proof of theorem 3.1 11

6. Mathematica scripts 12

7. Conclusion and future research 12

References 12

1. Definitions, Notations and Conventions

We now set the following notation, which remains fixed for the remainder of this paper:

Date: April 27, 2020.

2010Mathematics Subject Classification. 26E70, 05A30.

Key words and phrases. Dynamic equations on time scales, Partial differential equations on time scales, Partial dynamic equations on time scales, Partial differentiation on time scales, Dynamical systems .

1

• Let be a function f: T → R and t ∈ T^κ then f^∆(t) is delta time-scale deriva- tive [BP01] off

f^∆(t) = f(σ(t))−f(t)

µ(t) , µ(t)6= 0, whereµ(t) =σ(t)−t and σ(t)> t is forward jump operator.

• ∂f(t1, . . . , tn)

∆iti

, f_t^∆_iⁱ(t) is delta partial derivative of f: Λⁿ → R on n−dimensional time scale Λⁿ [BG04,AM02, Jac06], defined as a limit

f_t^∆_iⁱ(t) = lim

si→ti

s_i6=σi(ti)

f(t1, . . . , t_i−1, σi(ti), tt+1, . . . , tn)−f(t1, . . . , t_i−1, si, t_t+1, . . . , tn) σi(ti)−si

, whereσi(ti)> ti and σi(ti)−si 6= 0.

• Zis an integer time scale such that σ(t) =t+ 1 andµ(t) = 1.

• Ris a real time scale such that σ(t) =t+ ∆t and µ(t) = ∆t, ∆t →0.

• q^R is a quantum time scale such that σ(t) =qtand µ(t) = qt−t, [page 18 [BP01]].

• R^q is a quantum power time scale such that σ(t) =t^q and µ(t) =t^q−t.

• q^Rn is a quantum power time scale such thatσ(t) =qtⁿ> t, 0< q <1, µ(t) =qtⁿ−t and n is positive odd integer [AMT11].

• Dqf(x) is q−derivative [Jac09,Ern00, Ern08, KC01]

Dqf(x) = f(qx)−f(x)

qx−x , x6= 0, x∈R, q ∈R

• Dn,qf(t) is q−power derivative [AMT11]

Dn,qf(t) = f(qtⁿ)−f(t) qtⁿ−t , wheren is odd positive integer and 0< q <1.

• Dqf(x) is q−power derivative, the partial case of q−power derivative operator Dn,qf(x), q = 1

Dqf(x) = f(x^q)−f(x)

x^q−x , x^q 6=x, x∈R, q ∈R

• fm(x, b) is 2m+ 1−degree integer-valued polynomial [Kol16] in x, b, b≥1 fm(x, b) =P^m_b (x) =

b−1

X

k=0 m

X

r=0

A_m,rk^r(x−k)^r, x∈R, m∈N, whereA_m,r, m∈N is a real coefficient defined recursively

Am,r :=







(2r+ 1) ^2r_r

, if r =m,

(2r+ 1) ^2r_r Pm

d=2r+1A_{m,d 2r+1}^d ₍₋₁₎^d−1

d−r B2d−2r, if 0≤r < m,

0, if r <0 or r > m,

whereBt are Bernoulli numbers [Wei]. It is assumed that B₁ = ¹₂.

• hx−aiⁿ is powered Macaulay bracket [Mac19], hx−aiⁿ :=

((x−a)ⁿ, x≥a,

0, otherwise, a ∈Z

• {x−a}ⁿ is powered Macaulay bracket {x−a}ⁿ :=

((x−a)ⁿ, x > a,

0, otherwise, a∈Z

During the manuscript, the variableais reserved to be only the condition of Macaulay functions. If the power functionhx−aiⁿ or{x−a}ⁿ is written without parameter a e.g hxiⁿ or {x}ⁿ means that it is assumed a= 0.

2. Introduction

Time-scale calculus is quite graceful generalization and unification of the theory of dif- ferential equations. Firstly being introduced by Hilger [Hil88] in his Ph.D thesis in 1988 and thereafter greatly extended by Bohner and Peterson [BP01] in 2001, the calculus on time scales became a sharp tool in the world on differential equations. Various derivative operators like classical derivative _dx^df(x),q−derivative Dqf(x),q−power derivativeDqf(x), finite difference ∆f(x) etc, may be simply expressed in terms of time-scale derivative over particular time scale T. For instance,

f^′(x) =f^∆(x), x∈T:=R,

∆f(x) =f^∆(x), x∈T:=Z, Dn,qf(x) =f^∆(x), x∈T:=q^Rn, Dqf(x) =f^∆(x), x∈T:=q^R,

Dqf(x) = f^∆(x), x∈T:=R^q, . . . etc.

In context of Computer Science, namely object oriented programming paradigm, the time scale calculus may be thought as unified interface of derivative operator. Furthermore, the idea of time-scale calculus was slightly extended in [BHT17,BHT16,Cap09,MT09]. Let be a partial dynamic equation on time scales

∂fm(x, b)

∆x +∂fm(x, b)

∆b =αm(x, b)(x^2m+1)^∆,

where αm(x, b) is arbitrary differentiable function. In the next section, we inspect above partial dynamic equation, discussing its partial case for time scale Λ² = T₁×T₂ such that T₁ =T₂. In addition, a few polynomial identities are shown.

3. Main results

Theorem 3.1. Let fm(x, b) be a 2m+ 1−degree integer-valued polynomial. Let be a two- dimensional time scale Λ² =T₁×T₂ :={t = (x, b) : x∈ T₁, b∈T₂}. Let be T₁ =T₂. For every x, b ∈Λ², m∈N

∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x₀) = (x^2m+1)^∆(x0), where σ(x)> x is forward jump operator.

Theorem3.1 shows an identity between partial time-scale derivatives of fm(x, b) and ordi- nary time-scale derivative of odd power x^2m+1, x∈ R, m∈N. However, binomial theorem states that

x^t =X

r

t r

(x−1)^r =X

r

X

k

t r

r k

(−1)^r−kx^k =X

r

X

k

t k

t−k r−k

(−1)^r−kx^k (3.1)

=X

r

X

k

t k

t−k t−r

(−1)^r−kx^k

Above identity (3.1) is derived by means of binomial identity [eq. (4.1.9) [Gro16]] and further by symmetry of binomial coefficients. Therefore, theorem 3.1 may be rewritten in context of equation (3.1), e.g for every x, b∈Λ², m ∈N

∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x0) = X

r

X

k

t r

r k

(−1)^r−kx^k

!∆

(x0)

= X

r

X

k

t k

t−k r−k

(−1)^r−kx^k

!∆

(x₀)

= X

r

X

k

t k

t−k t−r

(−1)^r−kx^k

!∆

(x0) If we review the theorem3.1from another prospective, it opens an opportunity to connect its left part with discrete convolution of odd power x^2m+1, x∈ R, m∈ N. In [page 9 [Kol16],]

the following relations between odd power x^2m+1, x∈ R, m ∈N and discrete convolutions are stated

x^2m+1 =−1 +X

r

A_m,rhxi^r∗ hxi^r

x^2m+1 = 1 +X

r

A_m,r{x}^r∗ {x}^r

refer to the section definitions, notations and conventions 1 to find out the definitions of polynomials hxi^r, {x}^r. Hereof, the theorem 3.1 may express the relation between partial time-scale derivatives of fm(x, b) and ordinary time-scale derivative of discrete convolutions hxi^r∗ hxi^r, {x}^r∗ {x}^r, e.g

∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x₀) = X

r

Am,rhxi^r∗ hxi^r

!∆

(x0)

∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x₀) = X

r

Am,r{x}^r∗ {x}^r

!∆

(x0)

In order to express convolutionshxi^r∗ hxi^r, {x}^r∗ {x}^r in more convenient form, a two-sided Faulhaber-like formula [GMBn20] may be used, it is

x−1

X

k=1

k^m(x−k)^m = x^2m+1

(2m+ 1) ^2m_m + 2(−1)^m

m

X

k=0

m k

B_m+k+1 m+k+ 1x^m−k

Or, by identity in Riemann zeta function ζ(1−N) =−^B_N^N

x−1

X

k=1

k^m(x−k)^m = x^2m+1

(2m+ 1) ^2m_m −2(−1)^m

m

X

k=0

m k

ζ(−m−k)x^m−k, where BN are Bernoulli numbers.

4. Discussion and examples

To understand the nature of equation 3.1, let’s discuss an examples of some popular time scales, like integer time scale Z, real time scale R, quantum time scale q^R, quantum-power time scaleR^q. We use the principleDivide et Impera ! in order to understand entire behavior of theorem 3.1.

4.1. Time scale T=Z×Z.

Corollary 4.1. (Finite difference.) Let be a two-dimensional time scaleΛ² =Z×Z :={t= (x, b) : x∈Z, x∈Z}. For every x, b∈Λ², m ∈N

∆x^2m+1(x0) = ∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x0), where σ(x) =x+ 1.

Example 4.2. Let be x, b∈Λ² =Z×Z, m∈N and let m= 1, then

∂f₁(x, b)

∆x = 3b+ 3b²

∂f1(x, b)

∆x (t, σ(t)) = 3t+ 3t² And

∂f₁(x, b)

∆b = 1−6b²+ 6bx

∂f1(x, b)

∆b (t, t) = 1

Summing up above partial time-scale derivatives ^∂f1_∆x^(x,b)(t, σ(t)),^∂f1_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =Z×Z, m ∈N

∆x³(t) = ∂f1(x, b)

∆x (t, σ(t)) + ∂f1(x, b)

∆b (t, t) = 3t²+ 3t+ 1, which is ordinary finite difference ∆f(x).

Example 4.3. Let be x, b∈Λ² =Z×Z, m∈N and let m= 2, then

∂f₂(x, b)

∆x = 5b−30b²+ 40b³−15b⁴+ 10bx−30b²x+ 20b³x

∂f2(x, b)

∆x (t, σ(t)) = 5t+ 10t²+ 10t³+ 5t⁴

And

∂f₂(x, b)

∆b = 1 + 30b⁴−60b³x+ 30b²x²

∂f2(x, b)

∆b (t, t) = 1

Summing up above partial time-scale derivatives ^∂f2_∆x^(x,b)(t, σ(t)),^∂f2_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =Z×Z, m ∈N

∆x⁵(t) = ∂f₂(x, b)

∆x (t, σ(t)) + ∂f₂(x, b)

∆b (t, t) = 1 + 5t+ 10t²+ 10t³+ 5t⁴, which is ordinary finite difference ∆f(x).

Corollary 4.4. For every x, b∈Λ² =Z×Z, m∈N

∂fm(x, b)

∆x (t, σ(t)) =

2m

X

r=1

2m+ 1 r

t^r Corollary 4.5. For every x, b∈Λ² =Z×Z, m∈N

∂fm(x, b)

∆b (t, t) = 1 4.2. Time scale T=R×R.

Corollary 4.6. (Classical derivative.) Let be a two-dimensional time scale Λ² =R×R:=

{t= (x, b) : x∈R, b∈R}. For every x, b ∈Λ² =R×R, m ∈N d

dxx^2m+1(x0) = ∂fm(x, b)

∂x (x0, σ(x0)) + ∂fm(x, b)

∂b (x0, x₀), where σ(x) =x+ ∆x, ∆x→0.

Example 4.7. Let be x, b∈Λ² =R×R, m∈N and let m= 1, then

∂f₁(x, b)

∂x =−3b+ 3b²

∂f1(x, b)

∂x (t, σ(t)) =−3t+ 3t² And

∂f₁(x, b)

∂b = lim

∆b→0 6b−6b²+ 9∆b−18b∆b−14(∆b)²−3x+ 6bx+ 9x∆b

= 6b−6b²−3x+ 6bx,

∂f1(x, b)

∂b (t, t) = 3t

Summing up above partial time-scale derivatives ^∂f1_∂x^(x,b)(t, σ(t)),^∂f1_∂b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x∈Λ² =R×R, m∈N

d

dxx³(t) = ∂f1(x, b)

∂x (t, σ(t)) + ∂f1(x, b)

∂b (t, t) = 3t², which is classical derivative _dx^df(x).

Example 4.8. Let be x, b∈Λ² =R×R, m∈N and let m= 2, then

∂f₂(x, b)

∂x = lim

∆x→0 −15b²+ 30b³−15b⁴+ 5b∆x−15b²∆x+ 10b³∆x+ 10bx−30b²x+ 20b³x

=−15b²+ 30b³−15b⁴+ 10bx−30b²x+ 20b³x,

∂f2(x, b)

∂x (t, σ(t)) =−5t²+ 5t⁴ And

∂f₂(x, b)

∂b = lim

∆b→0(30b²−60b³+ 30b⁴+ 30b∆b−90b²∆b+ 60b³∆b

+ 10(∆b)² −60b(∆b)²+ 60b²(∆b)²−15(∆b)³+ 30b(∆b)³+ 6(∆b)⁴

−30bx+ 90b²x−60b³x−15x∆b+ 90bx∆b−90b²x∆b + 30(∆b)²x−60b(∆b)²x−15(∆b)³x+ 5x²−30bx² + 30b²x²−15x²∆b+ 30bx²∆b+ 10x²(∆b)²)

= 30b²−60b³+ 30b⁴−30bx+ 90b²x−60b³x+ 5x²−30bx²+ 30b²x²,

∂f2(x, b)

∂b (t, t) = 5t²

Summing up above partial time-scale derivatives ^∂f2_∂x^(x,b)(t, σ(t)),^∂f2_∂b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x∈Λ² =R×R, m∈N

d

dxx⁵(t) = ∂f2(x, b)

∂x (t, σ(t)) + ∂f2(x, b)

∂b (t, t) = 5t⁴, which is classical derivative _dx^df(x).

4.3. Time scale T=q^R×q^R.

Corollary 4.9. (Q-derivative [Jac09].) Let be a two-dimensional time scaleΛ² =q^R×q^R:=

{t= (x, b) : x∈q^R, b ∈q^R}. For every x, b∈Λ² =q^R×q^R, m∈N Dqx^2m+1(x₀) = ∂fm(x, b)

∆x (x₀, σ(x₀)) + ∂fm(x, b)

∆b (x₀, x₀), where σ(x) =qx, q > 1.

Example 4.10. Let be x, b∈Λ²=q^R×q^R, m∈N and let m= 1, then

∂f₁(x, b)

∆x (t, σ(t)) =−3qt+ 3q²t² And

∂f₁(x, b)

∆b = 3b−2b²+ 3bq−2b²q−2b²q²−3x+ 3bx+ 3bqx,

∂f1(x, b)

∆b (t, t) = 3qt+t²+qt²−2q²t²

Summing up above partial time-scale derivatives ^∂f1_∆x^(x,b)(t, σ(t)),^∂f1_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =q^R×q^R, m∈N

Dqx³(t) = ∂f1(x, b)

∆x (t, σ(t)) + ∂f1(x, b)

∆b (t, t) =t²+qt²+q²t²,

which is q−derivative Dqf(x), x∈Λ² =q^R×q^R.

For every x, b ∈ Λ² =q^R×q^R, m ∈N the following polynomial identity holds as q tends to zero

limq→0

∂f1(x, b)

∆b (t, t) = t²

However, for some m ∈N and x, b∈Λ² =q^R×q^R it would be concluded Corollary 4.11. For every x, b∈Λ² =q^R×q^R, m∈N

limq→0

∂fm(x, b)

∆b (t, t) =t^2m.

Example 4.12. Let be x, b∈Λ²=q^R×q^R, m∈N and let m= 2, then

∂f₂(x, b)

∆x =−15b²+ 30b³−15b⁴+ 5bx−15b²x+ 10b³x+ 5bqx−15b²qx+ 10b³qx,

∂f2(x, b)

∆x (t, σ(t)) = 5qt²−10q²t²−15q²t³+ 15q³t³+ 10q³t⁴−5q⁴t⁴ And

∂f₂(x, b)

∆b = 10b²−15b³+ 6b⁴+ 10b²q−15b³q+ 6b⁴q + 10b²q²−15b³q²+ 6b⁴q²−15b³q³+ 6b⁴q³

+ 6b⁴q⁴−15bx+ 30b²x−15b³x−15bqx+ 30b²qx

−15b³qx+ 30b²q²x−15b³q²x−15b³q³x+ 5x²

−15bx²+ 10b²x²−15bqx²+ 10b²qx² + 10b²q²x²,

∂f₂(x, b)

∆b (t, t) =−5qt²+ 10q²t²+ 15q²t³−15q³t³ +t⁴+qt⁴ +q²t⁴−9q³t⁴+ 6q⁴t⁴

Summing up above partial time-scale derivatives ^∂f2_∆x^(x,b)(t, σ(t)),^∂f2_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =q^R×q^R, m∈N

Dqx⁵(t) = ∂f₂(x, b)

∆x (t, σ(t)) + ∂f₂(x, b)

∆b (t, t) = t⁴+qt⁴+q²t⁴+q³t⁴+q⁴t⁴, which is q−derivative Dqf(x), x∈Λ² =q^R×q^R.

4.4. Time scale T=R^q×R^q.

Corollary 4.13. (Q-power derivative [AMT11].) Let be a two-dimensional time scale Λ² = R^q×R^q:={t= (x, b) : b∈R^q, x∈R^q}. For every x, b∈Λ² =R^q×R^q, m∈N

Dqx^2m+1(x0) = ∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x0), where σ(x) =x^q, q >1.

Example 4.14. Let be x, b∈Λ²=R^q×R^q, m∈N and let m= 1, then

∂f₁(x, b)

∆x (t, σ(t)) =−3t^q+ 3t^2q And

∂f₁(x, b)

∆b = 3b−2b² + 3b^q−2b^2q−2b^1+q−3x+ 3bx+ 3b^qx

∂f1(x, b)

∆b (t, t) =t²+ 3t^q−2t^2q+t^1+q

Summing up above partial time-scale derivatives ^∂f1_∆x^(x,b)(t, σ(t)),^∂f1_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =R^q×R^q, m∈N

Dqx³(t) = ∂f1(x, b)

∆x (t, σ(t)) + ∂f1(x, b)

∆b (t, t) =t²+t^2q+t^1+q, which is q−power derivativeDqf(x), x∈Λ² =R^q×R^q.

Example 4.15. Let be x, b∈Λ²=R^q×R^q, m∈N and let m= 2, then

∂f₂(x, b)

∆x =−15b²+ 30b³−15b⁴+ 5bx−15b²x+ 10b³x+ 5bx^q−15b²x^q+ 10b³x^q

∂f2(x, b)

∆x (t, σ(t)) =−10t^2q+ 15t^3q−5t^4q+ 5t^1+q−15t^1+2q+ 10t^1+3q And

∂f₂(x, b)

∆b = 10b²−15b³+ 6b⁴+ 10b^2q−15b^3q+ 6b^4q + 10b^1+q−15b^2+q+ 6b^3+q−15b^1+2q + 6b^2+2q+ 6b^1+3q−15bx+ 30b²x−15b³x

−15b^qx+ 30b^2qx−15b^3qx+ 30b^1+qx

−15b^2+qx−15b^1+2qx+ 5x²−15bx²+ 10b²x²

−15b^qx²+ 10b^2qx²+ 10b^1+qx²,

∂f2(x, b)

∆b (t, t) =t⁴+ 10t^2q−15t^3q+ 6t^4q−5t^1+q+t^3+q + 15t^1+2q+t^2+2q−9t^1+3q

Summing up above partial time-scale derivatives ^∂f2_∆x^(x,b)(t, σ(t)),^∂f2_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =R^q×R^q, m∈N

Dqx⁵(t) = ∂f2(x, b)

∆x (t, σ(t)) + ∂f2(x, b)

∆b (t, t) =t⁴+t^4q+t^3+q+t^2+2q+t^1+3q, which is q−power derivativeDqf(x), x∈Λ² =R^q×R^q.

Another polynomial identity, that is exponential sum holds Corollary 4.16. For every x, b∈Λ² =R^q×R^q, t∈R, m∈N

limq→0

∂fm(x, b)

∆b (t, t) =

2m

X

k=0

t^k

4.5. Time scale T=q^Rn ×q^Rn. In this subsection we discuss a pure quantum power time scaleq^Rj provided by Aldwoah, Malinowska and Torres in [AMT11], among with theq−power derivative operator Dn,qf(t) defined by

Dn,qf(t) = f(qtⁿ)−f(t) qtⁿ−t , where n is odd positive integer and 0< q <1.

Corollary 4.17. (Quantum power derivative[AMT11].) Let be a two-dimensional time scale Λ² =q^Rj ×q^Rj :={t= (x, b) : b ∈q^Rj, x∈q^Rj}. For every x, b ∈Λ² =q^Rj×q^Rj, m∈N

Dn,qx^2m+1(x0) = ∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x₀), where σ(x) =qtⁿ, σ(x)> x.

Example 4.18. Let be x, b∈Λ²=q^Rj ×q^Rj, m ∈N and let m = 1, then

∂f1(x, b)

∆x (x, b) =−3b+ 3b²

∂f₁(x, b)

∆x (t, σ(t)) = −3qt^j + 3q²t^2j And

∂f₁(x, b)

∆b = 3b−2b²+ 3b^jq−2b^1+jq−2b^2jq²−3x+ 3bx+ 3b^jqx

∂f₁(x, b)

∆b (t, t) =t²+ 3qt^j −2q²t^2j +qt^1+j

Summing up above partial time-scale derivatives ^∂f1_∆x^(x,b)(t, σ(t)),^∂f1_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =q^Rj×q^Rj, m∈N

Dn,qx³(t) = ∂f1(x, b)

∆x (t, σ(t)) + ∂f1(x, b)

∆b (t, t) =t² +q²t^2j +qt^1+j, which is q−power derivativeDn,qf(x), x∈Λ² =q^Rj ×q^Rj.

Another polynomial identity, that is exponential sum holds Corollary 4.19. For every x, b∈Λ² =q^Rj ×q^Rj, t ∈R, m ∈N

limj→0lim

q→1

∂fm(x, b)

∆b (t, t) =

2m

X

k=0

t^k An identity in even polynomials holds too

Corollary 4.20. For every x, b∈Λ² =q^Rj ×q^Rj, t ∈R, m ∈N limj→0lim

q→0

∂fm(x, b)

∆b (t, t) =t^2m

Example 4.21. Let be x, b∈Λ²=q^Rj ×q^Rj, m ∈N and let m = 2, then

∂f₂(x, b)

∆x (x, b) =−15b² + 30b³−15b⁴+ 5bx−15b²x+ 10b³x+ 5bqx^j −15b²qx^j + 10b³qx^j

∂f2(x, b)

∆x (t, σ(t)) =−10q²t^2j+ 15q³t^3j−5q⁴t^4j + 5qt^1+j −15q²t^1+2j+ 10q³t^1+3j

And

∂f2(x, b)

∆b = 10b²−15b³+ 6b⁴+ 10b^1+jq−15b^2+jq + 6b^3+jq+ 10b^2jq²−15b^1+2jq²

+ 6b^2+2jq²−15b^3jq³+ 6b^1+3jq³

+ 6b^4jq⁴−15bx+ 30b²x−15b³x−15b^jqx + 30b^1+jqx−15b^2+jqx+ 30b^2jq²x

−15b^1+2jq²x−15b^3jq³x+ 5x²−15bx² + 10b²x²−15b^jqx²+ 10b^1+jqx²+ 10b^2jq²x²,

∂f2(x, b)

∆b (t, t) =t⁴ + 10q²t^2j−15q³t^3j + 6q⁴t^4j

−5qt^1+j +qt^3+j + 15q²t^1+2j +q²t^2+2j

−9q³t^1+3j

Summing up above partial time-scale derivatives ^∂f1_∆x^(x,b)(t, σ(t)),^∂f1_∆b^(x,b)(t, t) in point t ∈ R, we get time scale derivative of odd power x^2m+1, x, b∈Λ² =q^Rj×q^Rj, m∈N

Dn,qx⁵(t) = ∂f₁(x, b)

∆x (t, σ(t)) + ∂f₁(x, b)

∆b (t, t) =t⁴+q⁴t^4j+qt^3+j +q²t^2+2j+q³t^1+3j, which is q−power derivativeDn,qf(x), x∈Λ² =q^Rj ×q^Rj.

5. Proof of main theorem By [Lemma 3.1 [Kol16]], for every x∈R, m∈N it is true that

fm(x, x) =x^2m+1 (5.1)

Proof of theorem 3.1. Let be x, b ∈ Λ² =T₁×T₂ := {t= (x, b) : x ∈ T₁, b ∈T₂}. Let be T₁ =T₂, then

(x^2m+1)^∆= lim

b→xlim

t→x

fm(σ(x), σ(b))−fm(t, b)

σ(x)−t , (5.2)

where σ(x) > x is forward jump operator. However, equation (5.2) is not a timescale derivative of fm(x, b) over xhow it might seem because of denominator σ(x)−t. Parameter b of fm(x, b) is implicitly incremented as well. Let’s try to express nominator of (5.2) in terms of partial derivative ^∂fm_∆b^(x,b) on timescales. Let be the following equation

fm(x, σ(b))−fm(x, b) =fm(x, b)^∆_b ·∆b Assume that nominator of (5.2) equals to, ast →x

fm(σ(x), σ(b))−fm(x, b) =fm(x, σ(b))−fm(x, b) +A

whereA is yet implicit term. Let’s now collapse the terms fm(x, b) from both sides of above equation, such that

fm(σ(x), σ(b)) =fm(x, σ(b)) +A Therefore,

A=fm(σ(x), σ(b))−fm(x, σ(b)) =fm(x, b)^∆_x(x, σ(b))·∆x

Now, let’s express the nominator of (5.2) as follows

fm(σ(x), σ(b))−fm(x, b) =fm(x, b)^∆_x(x, σ(b))·∆x+fm(x, b)^∆_b (x, b)·∆b

fm(σ(x), σ(b))−fm(x, b) =fm(x, b)^∆_x(x, σ(b))·(σ(x)−x) +fm(x, b)^∆_b (x, b)·(σ(b)−b) We can collapse the terms (σ(x)−x), (σ(b)−b) in above expressions, as b→x. Therefore,

fm(σ(x), σ(x))−fm(x, x)

σ(x)−x =fm(x, b)^∆_x(x, σ(x)) +fm(x, b)^∆_b (x, x)

Finally, by the identity (5.1) we can express timescale derivative of x^2m+1, x ∈ Λ² = T₁ × T₂, m∈N as

∂fm(x, b)

∆x (x0, σ(x0)) + ∂fm(x, b)

∆b (x0, x0) = (x^2m+1)^∆(x0).

This completes the proof.

6. Mathematica scripts

To fulfill our study, we attach here a link to the set of Mathematica programs, designed to verify the results of current manuscript. To reach these programs follow the link [Kol20].

7. Conclusion and future research

In this manuscript we have discussed partial time scale differential equation involving derivatives of polynomials in context of time scale Λ² = T₁ ×T₂ where T₁ = T₂. Future research can be conducted to study the case T₁ 6= T₂, which makes the theorem 3.1 to be generalised

∂fm(x, b)

∆x +∂fm(x, b)

∆b =αm(x, b)(x^2m+1)^∆,

whereαm(x, b) is arbitrary differentiable function. Also, it is worth to discuss the theorem3.1 in context of high order derivatives on time scales. We have established a few power identities, and shown the theorem 3.1 for different 2-dimensional time scales Λ² like integer time scale Z×Z, real time scale R×R, quantum time scale q^R×q^R and quantum power time scale R^q×R^q.

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E-mail address: kolosovp94@gmail.com URL:https://kolosovpetro.github.io