On the Cauchy problem for multidimensional diﬀerence equations in rational cone

Nekrasova Tatiana I.

Multidimensional Difference

УДК 517.55

On the Cauchy Problem for Equations in Rational Cone

Tatiana I. Nekrasova*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 10.02.2015, received in revised form 15.03.2015, accepted 28.04.2015 The Cauchy problem for multidimensional difference equations in rational cone is formulated and sufficient condition for its solvability is given. The notion of multisection of multiple Laurent series with the support in a rational cone is defined. The formulae which express any multisection through original series are presented.

Keywords: Cauchy problem, rational cone, generating function, multisection.

Introduction

In this paper we discuss some issues related to the Cauchy problem for multidimensional difference equations whose solutions are sought at the intersection of rational cone K with integer lattice. Methods of the theory of generating functions (z-transformations) play an important role in the study of the Cauchy problem. Problems of solvability of the Cauchy problem in the positive octant of the integer lattice and the algebraic nature of the generating function of its solution are studied in [1]. When passing from positive octant to more general case of a rational cone difficulties arise. They are associated with the fact that the cone K, in general, not unimodular.

In the first section we formulate the Cauchy problem and provide the sufficient condition for its solvability (see Theorem 1). The multi-dimensional analogue of the notion of the multisection of a power series helps us to overcome mentioned above difficulties in study of generating functions (series) with supports in rational cones. This multi-dimensional analogue is defined in the second part of the paper. Relation that represents the multisections of the series in terms of the original series (see Theorem 2) is also presented in the second part of the paper.

1. On solvability of the Cauchy problem

Let us introduce complex-valued functions f(x) = f(xi,...,xn) of integer variables ii,...,in. We define the shift operators Sj with respect to the variables xj: Sjf(x) = f (xi,..., xj_i, xj + 1, xj+i,..., xn) and polynomial difference operator of the form

P(S)= £ Sw,

wen

where l C Zn is a finite set of points of n-dimensional integer lattice Zn, 5" = 5"1.....5%™ and

cw are constant coefficients of the difference operator. Let us consider the difference equation of the form

P(5)f (x) = g(x),x G X, (1)

where f (x) is unknown function and g(x) is a function defined on some fixed set X C Zn. Subset X0 C X is called the initial (boundary) set. Let us formulate the problem. Find function f (x) that satisfies equation (1) and the following equation

f (x) = p(x),x G Xo (2)

for a given function y(x).

It is naturally to designate this problem as Cauchy problem for equation (1) and function y(x) in condition (2) is designated as initial data of the Cauchy problem. The existence and uniqueness of solutions of the Cauchy problem depends on all objects involved in the setting this problem: the difference operator P( delta), set X on which we define function g(x) in equation (1) and set X0 on which we define the initial data <^(x).

We are interested in problems of the form (1)-(2) that arise in combinatorial analysis. Dif-

m

ference operator in the one-dimensional case (see [2,3]) is given by P(5) = ^ cu5U, cm = 0,

the set X is the set of non-negative integers Z+ and the set X0 = {0,1,... ,m — 1}. Under these conditions problem (1)-(2) obviously has the unique solution. In multidimensional case we usually have X = Z+ and the choice of set X0 depends on the properties of the set l on which the characteristic polynomial P is defined. The problem of correct formulation (formulation that ensures the existence and uniqueness of the solution) of Cauchy problem in the positive octant Z+ of the integer lattice for difference equation (1) was studied [1]. In addition, there was studied the algebraic nature of the generating function of solution of the difference equation.

In this paper we study the problem of solvability of Cauchy problem (1)-(2), that is, the problem of existence and uniqueness of the function f (x) that satisfies (1)-(2). The function is defined at integer points K n Zn of rational cone K. Let us give some needed notations and definitions.

Let us assume that a1,..., an are linearly independent vectors with integer coordinates aj = (aj,..., a^), aj G Z. Rational cone generated by the vectors a1,..., an is the set K = {x G Rn : x = A1a1 + • • • + Anan, Aj G R+, j = 1,..., n}. Let us note that this cone is simplicial cone, that is, each element of the cone is uniquely expressed in terms of generators. In addition, simplicial cone is also salient cone, that is, this cone contains no lines. Let us introduce matrix A. The columns of this matrix are composed of the vectors aj and A = det A. If A = 1 then the cone K is a unimodular cone.

Let us define a partial order > between points u, v G Rn as follows

k

u ^ v ^ u G v + K,

K

where v+K is the shift of the cone K by the vector v. In addition, we write u ^ v if u G K\{v+K}.

K

Let us fix m G l and specify problem (1)-(2) as follows: we take the intersection K n Zn of rational cone and the integer lattice as X and X0 = {x G K n Zn : x ^ m}. Let us find a

K

function f(x) that satisfies the equation

P(5)f (x) = g(x),x G K n Zn (3)

and coincides with the given function <^(x) on set X0:

f (x) = y(x),x G Xo. (4)

In the positive octant Z+ of integer lattice (that is Aj = ej, ej — unit vectors, j = 1,..., n) degrees of monomials zx on the set of variables are defined as follows xi + x2 + dots + xn and degrees of the monomials are the same when their exponents lie on the hyperplane xi + x2 + • • • + xn = d. Note that v = ei + e2 + • • • + en is normal vector to this hyperplane. In the case of an arbitrary simplicial rational cone generated by the vectors ai, a2,..., an it is natural to take v = ai + a2 + • • • + an and denote (v, x) = vixi + • • • + vnxn.

Theorem 1. If for any w G 0 the condition (v, w — m) ^ 0 is fullfield and m is the only point of 0, which lies on the hyperplane (v, xm) = 0, then problem (3)-(4) has the unique solution.

The proof of theorem 1 is reduced to the solvability of an infinite system of linear equations with an infinite number of variables. The system has a feature: each equation contains only a finite number of unknowns. Such system is consistent, if any system of a finite number of equations is consistent (see, [4], Ch. 6, Lemma 6.3.7). We construct sequence of subsystems (3)-(4) that consists of a finite number of equations. These subsystems are arranged so that each following subsystem includes all equations of the previous one. Because of the mentioned above lemma the compatibility of each of these subsystems consistency means that system (3)-(4) is consistent.

Let us introduce the relation -< on the lattice points of rational cone K. If -< is the relation

K

of lexicographical order in Z+ then for x, y G K n Zn we define the ratio K as follows

x -< y ^ AA-ix -< AA-iy,

K

where A_i is the matrix inverse to A and A = det A.

For the vector v we consider the linear in x function (v, x), x G K. We form the set of its values on the points of the set K n Zn in orderly pattern and designate it as S. Note that S C Z+ because v is in the dual cone to cone K. We defined weighted lexicographic ordering < on the set of lattice points of the cone K as follows. For x, y G K n Zn the ratio x < y means that (v, x) < (v, y) and if (v, x) = (v, y) then x K y.

Let us take some s G S. Unknown elements of the set are numbered by Js = {y G K n Zn : (v, y) < s} and equations are numbered by elements of two sets Is = {x G K n Zn : (v, x) < s — (v, m)} and Im,s = {m G X0 : (v, < s}. Points x of the set Is « » are assigned numbers of points m + x ¿nJs. Points x of the set Is are numbered in the same way as points m + x G Js. We denote number of elements of a finite set M by #M then it is easy to see that #Is +#Im,s = #Js. Thus, we obtain a system of linear equations for the unknown f (y), y G Js of the form

y^ cwf (x + w) = g(x),x G Is, (5)

wen

f (m) = G Im,s. (6)

Determinant of system (5)-(6) is denoted by Am, s.

Proof of Theorem 1.

All elements in the rows of the determinant Am,s but one are equal to zero. This element is equal to one and it lies on the main diagonal. This follows from the algorithm of ordering of

unknowns and equations of system (5)-(6). Consider the rows of the determinant corresponding to equations (5). Firstly, only are equal to zero. Secondly, it follows from the conditions of Theorem 1 that w < m, w £ Q, w = m then x + w < x + m. So in the rows of the determinant that corresponds to equation (5), the last non-zero element is cm. Element cm stands on the main diagonal because of the equation has the number x + m and number of unknown is y = x + m. Thus, Am s is the determinant of a lower triangular matrix. Non-zero elements cm are on the main diagonal of the matrix, that is, Am s = 0 for all s £ S. □

Note that in the case K = R+ theorem 1 is proved with the use of other method (see [1]).

2. Multisection of Laurent series with the support in a rational cone

We recall the notion of multisection of power series in the one-dimensional case. For a fixed

positive integer q we define the k-th q-section Tkjq of series G(£) = as follows

j=0

G(0 = £ g(k + jq)£fc+jq, k = 0, 1, . . . , q - 1,

j=0

where Tkiq is a linear operator acting on the ring of formal power series C[[£]]. It is known that every k-th q-section of series is expressed through the original series (see [5]) as follows

1 q

G(£) = -V rq-kjG(rje),k = 0,1,..., q - 1, (7)

q

where r is a primitive q-th root of unity, that is rq = 1, r =1.

Let us note that multisection is used to prove identities with binomial coefficients and the Bernoulli numbers [5]. The need for a multi-dimensional analogue of the notion of multisection multiple series arises in the study of the Cauchy problem for multidimensional difference equations (see [1,6-9]). In particular, this is the case when supports of generating functions of equation solutions is in rational cone. Generating functions are naturally divided into the sum of multisections. The question arises as to whether the original series and multisections are of the same class, for example, the class of rational or algebraic functions. Let us introduce some needed notations and definitions.

Let A = {x £ Z" : x = Aiax + .. .+A„a", Aj £ Z, i = 1,..., n} be sublattice of Zn generated by the vectors a1,..., a". We fix t £ AnK and introduce ^ = (^1,..., ), where are coordinates

of t in the basis a1,..., a". We denote by nT the parallelotope nT = {x £ R" : 0 < x < t} and

k k

denote by AT = {x £ Z" : x = A1^1a1 + ... + An^na", Aj £ Z, i = 1,..., n} the sublattice of Z" generated by the vectors ^1a1,..., ^1a". Next, we assume that v are points with integer coordinates that belong to parallelotope nT. The number of points is equal to the volume of

parallelotope Vo1(nT) = ...^"A". It is obviously that |J (v + AT) = Z"

cenTnzn

Let CK [[z]] be the ring of Laurent series of the form F(z) = f . Then, the v-th

xEKnZ"

t-section of multiple Laurent series F(z) is a series of the form

f(x)~x

xeKn|v+AT }

Tv,t F (z) = \ f (x)zx. (8)

It is easy to see that any series F(z) from the ring CK [[z]] can be uniquely expressed as the

sum

F (z) = Tv,t F (z). (9)

cenTnzn

The following theorem generalizes relation (7) to the case of multiple series.

Theorem 2. Every v-th t-section Tv,TF(z) is expressed in terms of the original series F(z) as follows

Tv tF(z) = -V RT-v0JF(RJz), (10)

where R = (Ri,..., Rn), Rj = 1, j = 1,..., n is some solution of the system of equations

= 1, i = 1,...,n, (11)

and J = (ji,..., jn), 1 ^ ji ^ Mi^, ■ ■ ■, 1 ^ ^ MnA, where Mi are coordinates of t in the basis a1,..., an.

From (10) we immediately get

Corollary 1 If the series F(z) is a rational (algebraic) series then the series Tv,TF(z) for any t e A n K and v e nT is rational (algebraic) series.

Let us consider first the case K = R+, that is, the case of multiple it power series. Let

£ = (£i,...,£n) e Cn, = ...... We fix q = (qi,...,q„) e Z+ and consider the

half-open parallelepiped nq = {x e R+ : 0 ^ x < q}. Number of points k e nq n Zn with integer coordinates is equal to #nq n Zn = qi •... • qn. Integer lattice Zn can be written as Zn =

|J (k + qZn), where the union is taken over all shifts of sublattice qZn = (qiZ) x • • • x (qnZ)

ten,nzn

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by vectors k e nq n Zn. Let us denote

F(£)= £ h(A)£j e C[[£]]. (12)

Then from (8) we obtain F (£) = £ h(A)£j.

A£fc+qZ+

Theorem 3. For k-th q-section of the power series (12) the following relation

TM F (£) =-1-V rq-fc0J F (rJ £), (13)

qi • ... • qn J

is valid, where J = (ji,..., jn), 1 ^ ji ^ qi, ■ ■ ■, 1 ^ jn ^ qn, r = (ri,..., rn) and rj is a primitive qj-th root of unity.

Proof. It is easy to verify that the k-th q-section of the power series (12) can be written as

TU F (£) = V h(ji,..., ki + qi ji,..., jn) j ... ji... j. (14)

jez+

Note that the last equality can be taken as a definition of multisection. Sequential execution (composition) of qi-th and qj-th sections is designated as T|,q, o qj. It should be noted that operation o is commutative and associative.

For k = (ki,..., kn) and q = (qi,..., qn) the k-th q-section of multiple power series F(£) is the following series

f(о = Ti^ ◦ • • • ◦ ît„,,„f(e) = £ h(k + qj)ek+qi.

jez+

Using (14) and n times (7), we obtain

f (e) = t^ о •••о F (e) =

tu◦•••◦ ^(j i,...,jn-i,kn+qnjn)^^1 ...en-enn+qе

jez+

= Tli,qi О • • • О îkrr_1iiqn_i ГГfcnjn F (Ci, . . . , in- i , j Én) =

qn jn=0

, qn-1

= T i ◦ ◦ Tn-2 1 V"^ rqn-i-kn-i jn-i v

= Tfci,qi ◦•••◦ T kn-2 ,qn-2 q Z^ rn- i v

qn- i - n

1 qn

v - X) rnn-knjn f (ei,...,en-2,rjn-1 en- i ,rjn en ) =

qn jn = 0

1 qi i qn-i i qn

.i ^ r qi-klj1 ... £ rnrii-kn-ijn-1 _L ^ rnn-k»i"F(rj1 e i,..., rjn-1 en- i, rjnen)

q i ji=0 qn- i jn-i=0 qnjn=0

1 X ^ q—k^ 7 J

(r

qi • ... • qn J

Erq-k0J f (rj e).

□

Now we apply (13) for the proof of Theorem 2.

Proof of the Theorem 2. We show that (10) is equivalent to (13) after some transformations. On the left hand side of (10) we introduce the change of variables zA = £A, where zA =

(z i1 ... zTn,..., z а ... znn). Let us denote k = ДА i v, q = ДА 1t and note that q = Д^.

After change x = —- of summation index x G K П Zn on A G |J (t + Д2+), where

Д tenA nzn

n

Пд = {y G Rn : 0<y < (Д,..., Д)}, we obtain

же{«+лт}nK Aez+

Next, we denote

f ( дAA), if A G U (t +

h(A) = < ieninZn

*o, if a G U (t + Д2+).

tenAnzn

Then relation (15) becomes

Aez+

A£Z+

Now we do the same change of variables zA = on the right hand side of (10)

1

Mi • • -MnA'

Y, RT-v0JF(RJz) =

j

]TR i i Ak0J f (_L AA)RJ i AAz i AA =

J Ae U (i+AZ+)

tenA nzn

■£rq-fc0j £ f (_A AA)rjA eA =

^ J Ae u (i+Az+)

tenAnzn

= —1— vr«-fcoJ v h(A)rjAeA = —1— £r«-fcoJf(rje).

Thus, by theorem 2 we find that the right hand side is equal to the left hand side

tm F (e) = —£ rq-fc0j F (rj e).

q1 . . . qn J

After returning to the variable z we find that relation (10) is valid. □

References

[1] M.Bousquet-Melou, M.Petkovsek, Linear recurrences with constant coefficients: the multivariate case, Discrete Mathematics, 225(2000), 51-75.

[2] R.F.Isaacs, A Finite Difference Function Theory, Univ. Nac. Tucuman. Revista A., 2(1941), 177-201.

[3] A.O.Gelfond, The calculus of finite differences Moscow, KomKniga, 2006 (in Russian).

[4] L.Hormander, An introduction to complex analysis in several variables, 3 Ed., North Holland, 1990.

[5] J.Riordan, Combinatorial Identities, John Wiley and Sons, New-York, 1968.

[6] E.K.Leinartas, Multiple Laurent series and fundamental solutions of linear difference equations, Siberian Math. J, 48(2007), no. 2, 335-340.

[7] E.K.Leinartas, The stability of the Cauchy problem for the homogeneous difference operator and the amoeba of characteristic set, Siberian Math. J., 52(2011), no. 5, 1087-1095.

[8] T.I.Nekrasova, Cauchy problem for multidimensional difference equations in lattice cone, Journal of Siberian Federal University. Mathematics & Physics, 5(2012), no. 4, 576-580 (in Russian).

[9] T.I.Nekrasova, Sufficient conditions of algebraicity of generating functions of the solutions of multidimensional difference equations, Izvestiya Irkutskogo Gosudarstvennogo Universiteta, 6(2013), no. 3, 88-96 (in Russian).

1

О задаче Коши для многомерных разностных уравнений в рациональных конусах

Татьяна И. Некрасова

Сформулирована задача Коши для многомерных 'разностных уравнений в рациональных конусах, дано достаточное условие ее разрешимости. Определено понятие мультисекции кратных рядов Лорана с носителями, лежащими в рациональных конусах, и приведена формула, выражающая всякую мультисекцию через исходный ряд.

Ключевые слова: задача Коши, рациональный конус, производящая функция, мультисекция.

On the Cauchy problem for multidimensional diﬀerence equations in rational cone Текст научной статьи по специальности «Математика»

Аннотация научной статьи по математике, автор научной работы — Nekrasova Tatiana I.

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