Functions satisfying
holonomic q-differential equations
Wolfram Koepf
Department of Mathematics and Computer Science,
University of Kassel, Germany,
E-mail: [email protected]
Predrag M. Rajković
Department of Mathematics, Faculty of Mechanical Engineering
E-mail: [email protected]
Sladjana D. Marinković
Department of Mathematics, Faculty of Electronic Engineering
E-mail: [email protected]
University of Niš, Serbia
Abstract. In a similar manner as in the papers [7] and [8], where explicit algorithms
for finding the differential equations satisfied by holonomic functions were given, in
this paper we deal with the space of the q-holonomic functions which are the solutions
of linear q-differential equations with polynomial coefficients. The sum, product and
the composition with power functions of q-holonomic functions are also q-holonomic
and the resulting q-differential equations can be computed algorithmically.
Mathematics Subject Classification: 39A13, 33D15
Key words: q-derivative, q-differential equation, algorithm, algebra of q-holonomic
functions.
1
Preliminaries
The purpose of this paper is to continue the research exposed in [7] and [8]. There, the
authors discussed holonomic functions which are the solutions of homogeneous linear
differential equations with polynomial coefficients.
In the present investigation, we consider a similar problem from the point of view
of q-calculus. As general references for q-calculus see [2] and [4]. We begin with a
few definitions.
Let q 6= 1. The q-complex number [a]q is given by
[a]q :=
1 − qa
,
1−q
a ∈ C.
Of course
lim [a]q = a .
q→1
The q-factorial of a positive integer [n]q and the q-binomial coefficient are defined by
n
[n]q !
[0]q ! := 1, [n]q ! := [n]q [n − 1]q · · · [1]q ,
=
.
k q
[k]q ![n − k]q !
1
The q-Pochammer symbol is given as
(a; q)0 = 1 ,
(a; q)k = (1 − a)(1 − aq)(1 − aq 2 ) · · · (1 − aq k−1 ),
2
(a; q)∞ = lim (1 − a)(1 − aq)(1 − aq ) · · · (1 − aq
k = 1, 2, . . . ,
k−1
k→∞
)
and
(a; q)∞
(|q| < 1, λ ∈ C).
(aq λ ; q)∞
The q-derivative of a function f (x) is defined by
(a; q)λ =
Dq f (x) :=
f (x) − f (qx)
(x 6= 0),
x − qx
Dq f (0) := lim Dq f (x),
x→0
(1)
and higher order q-derivatives are defined recursively
Dq0 f := f,
Dqn f := Dq Dqn−1 f,
n = 1, 2, 3, . . . .
(2)
Of course, if f is differentiable at x, then
lim Dq f (x) = f 0 (x) .
q→1
The next four lemmas are well-known in q-calculus and their proofs can be seen,
for example, in [3] or [4].
Lemma 1.1. For an arbitrary pair of functions u(x) and v(x) and constants α, β ∈
C and q 6= 1, we have linearity and product rules
Dq αu(x) + βv(x) = αDq u(x) + βDq v(x),
Dq u(x) · v(x) = u(qx)Dq v(x) + v(x)Dq u(x)
= u(x)Dq v(x) + v(qx)Dq u(x) .
Lemma 1.2. The Leibniz rule for the higher order q-derivatives of a product of functions is given as
n X
n
Dqn u(x) · v(x) =
Dn−k u(q k x) Dqk v(x).
k q q
k=0
Lemma 1.3. For an arbitrary function u(x) and for t(x) = cxk (c, k ∈ C, q k 6= 1)
we have for the composition with t(x)
Dq (u ◦ t)(x) = Dqk u(t) · Dq t(x).
Lemma 1.4. The values of the function for the shifted argument and for higher qderivatives are connected by the two relations:
n
X
k
n
(−1)k (1 − q)k
q (2) xk Dqk f (x),
(3)
f (q n x) =
k q
k=0
n
X
k
1
k n
Dqn f (x) =
(−1)
q (2)−(n−1)k f (q k x).
(4)
(1 − q)n xn
k q
k=0
2
For our further work, it is useful to write the product rule in slightly different form.
Lemma 1.5. The product rule for the q-derivative can be written in the form
Dq u(x) · v(x) = u(x)Dq v(x) + v(x)Dq u(x) − (1 − q)xDq u(x)Dq v(x) .
(5)
In the same manner, higher q-derivatives can be expressed by
n X
n
X
Dqn u(x) · v(x) =
ανµ (x)Dqν u(x)Dqµ v(x) ,
ν=0 µ=0
where ανµ (x) are appropriate polynomials.
Let us finally recall that the q-hypergeometric series is given by ([2], [6])
∞ Qr
1+s−r
X
xk a1 , a2 , . . . , ar j=1 (aj ; q)k
k (k
2)
Q
q,
x
:=
(−1)
q
.
r φs
s
b1 , b 2 , . . . , b s j=1 (bj ; q)k (q; q)k
k=0
2
On q-holonomic functions
For every function f (x) which is a solution of a polynomial homogeneous linear qdifferential equation
n
X
p̃k (x) Dqk f (x) = 0
(p̃n 6≡ 0)
(p̃k ∈ K(q)[x], n ∈ N)
(6)
k=0
we say that f (x) is a q-holonomic function. The smallest such n is called the
holonomic order of f (x). Here K is a field, typically K = Q(a1 , a2 , . . .) or K =
C(a1 , a2 , . . .) where a1 , a2 , . . . denote some parameters. An equation of type (6) is
called a q-holonomic equation.
Example 2.1. Since
Dq xs = [s]q xs−1 (x, α, s ∈ R),
we have
f (x) = xs ⇒ xDq f (x) − [s]q f (x) = 0 ,
or
(q − 1) x Dq f (x) − (q s − 1) f (x) = 0 ,
i.e. the power function is (for integer s) a q-holonomic function of first order.
Example 2.2. For 0 < |q| < 1, λ ∈ R, x 6= 0, 1, we have
−[λ]q
(x; q)λ .
Dq (x; q)λ = −[λ]q (qx; q)λ−1 =
1−x
Hence
f (x) = (x; q)λ ⇒ (x − 1)Dq f (x) − [λ]q f (x) = 0
3
or
(q − 1) (x − 1) Dq f (x) − (q λ − 1) f (x) = 0 .
Therefore the q-Pochhammer symbol is (for integer λ) also q-holonomic of first order.
Similarly, from
1
1
Dq (x; q)∞ = −(1 − q)−1 (qx; q)∞ = −
(x; q)∞ ,
1−q1−x
we get
f (x) = (x; q)∞
⇒
(1 − x)Dq f (x) +
1
f (x) = 0.
1−q
Example 2.3. The small q-exponential function
eq (x) = 1 φ0
0
−
X
∞
1
q, x =
xn ,
(q;
q)
n
n=0
|x| < 1, 0 < |q| < 1 ,
has q-derivative
eq (x) − eq (qx)
x − qx
!
∞
∞
X
X
1
1
1
n
n
x −
(qx)
=
x − qx n=0 (q; q)n
(q; q)n
n=0
Dq eq (x) =
∞
X
xn − (qx)n
1
x − qx n=0 (q; q)n
(
)
∞
X
1
1 − qn
=
x+
xn
2 ) · · · (1 − q n−1 )(1 − q n )
x − qx
(1
−
q)(1
−
q
n=2
(
)
∞
X
x
1
k
=
1+
x
x − qx
(1 − q)(1 − q 2 ) · · · (1 − q k )
=
k=1
1
=
eq (x),
1−q
i.e. the small q-exponential function is q-holonomic of first order:
f (x) = eq (x) ⇒ (1 − q)Dq f (x) − f (x) = 0.
Note that this q-differential equation as well the resulting q-differential equations of the
next four examples and similar ones can be obtained completely automatically by the
qsumdiffeq command of the Maple package qsum by Böing and Koepf [1]. The
above equation, e.g., is obtained using the command
qsumdiffeq(1/qpochhammer(q,q,n)*xˆn,q,n,f(x))}
4
Example 2.4. The big q-exponential function
Eq (x) = 0 φ0
−
−
n
X
∞
q( 2 ) n
q, −x =
x ,
(q; q)n
n=0
0 < |q| < 1
has q-derivative
n
n
∞
∞
X
q( 2 ) n X q( 2 )
x −
(qx)n
(q;
q)
(q;
q)
n
n
n=0
n=0
1
Dq Eq (x) =
x − qx
!
=
1
Eq (qx).
1−q
which can obtained be in a similar way as in Example 2.3. Since
f (qx) = f (x) − (1 − q)x(Dq f )(x),
we conclude that the big q-exponential function is also q-holonomic of first order:
f (x) = Eq (x) ⇒ (1 − q)(x + 1)Dq f (x) − f (x) = 0.
Example 2.5. For 0 < |q| < 1, q-sine and q-cosine functions
sinq (x) =
∞
eq (ix) − eq (−ix) X (−1)n
=
x2n+1 ,
2i
(q;
q)
2n+1
n=0
∞
eq (ix) + eq (−ix) X (−1)n 2n
=
cosq (x) =
x ,
2
(q; q)2n
n=0
satisfy
(1 − q)2 Dq2 f (x) + f (x) = 0
and are therefore q-holonomic of second order.
Example 2.6. The q-hypergeometric series r φs is q-holonomic. The qsumdiffeq
command computes in particular for
a, b f (x) = 2 φ1
q, x
c the q-holonomic equation
0
=
(xabq − c)x(q − 1)2 Dq2 f (x)
+(−xb − xa + 1 + xabq − c + xab)(q − 1)Dq f (x)
+(−1 + a)(−1 + b)f (x) .
Example 2.7 Many q-orthogonal polynomials are q-holonomic. The Big q-Jacobi polynomials (see e.g. [5], 3.5) are given by
−n
q , abq n+1 , x f (x) = Pn (x; a, b, c; q) = 3 φ2
q, q .
aq, cq
5
They satisfy the q-holonomic equation
0 = q n a(bqx − c)(q − 1)2 (1 − qx) Dq2 f (x)
+(q−1)(abq n+1 +abq 2n+1 x+x−q n a−q n c−abq n+1 x−abq n+2 x+q n+1 ac)Dq f (x)
+(q n − 1)(abq n+1 − 1)f (x)
which is again easily determined by the qsumdiffeq command.
The following lemma will be the crucial tool for the investigations of the next section.
Lemma 2.1. If f (x) is a function satisfying a holonomic equation (6) of order n, then
the functions Dql f (x) (l = n, n + 1, . . .) can be expressed as
Dql f (x) =
n−1
X
(l)
pk (x)Dqk f (x),
(7)
k=0
(l)
where pk (x) are rational functions defined by

δkl ,
0≤l <n−1,







p̃k (x)
(l)
−
,
l=n
pk (x) =
p̃

n (x)





 (l−1)
(l−1)
(l−1)
(n)
pk−1 (qx) + Dq pk
(x) + pn−1 (qx)pk (x) ,
l>n,
for 0 ≤ k ≤ n − 1 and 0 for other k’s.
Proof. The representations (7) and the corresponding coefficients are evident by Equation (6) for l = 0, 1, . . . , n. By q-deriving and using Lemma 1.1, from
Dqn f (x) =
n−1
X
(n)
pk (x)Dqk f (x)
k=0
we get
Dqn+1 f (x)
=
n−1
X
(n)
Dq pk (x)Dqk f (x)
k=0
=
n−1
X
(n)
pk (qx)Dqk+1 f (x) +
k=0
=
n−1
X
=
(n)
Dq pk (x) Dqk f (x)
k=0
(n)
(n)
(n)
pk−1 (qx) + Dq pk (x) Dqk f (x) + pn−1 (x)Dqn f (x)
k=0
n−1
X
n−1
X
(n+1)
pk
(x)Dqk f (x),
k=0
6
with
(n+1)
pk
(n)
(n)
(n)
(n)
(x) = pk−1 (qx) + Dq pk (x) + pn−1 (qx)pk (x)
(0 ≤ k ≤ n − 1).
Repeating the procedure, we get the representation and coefficients for arbitrary l > n.
♦
We finish this section by noting that there are functions which are not q-holonomic.
Lemma 2.2. The exponential function f (x) = ax (a > 0, a 6= 1) is not q-holonomic.
Proof. Taking successive q-derivatives of f (x) := ax generates iteratively the functions
2
of the list L := {ax , aqx , aq x , . . .}. Since the members of L are linearly independent
over K(q)[x], the linear space over K(q)[x] generated by L has infinite dimension. This
is equivalent to the fact that there is no q-holonomic equation for f (x). ♦
3
Operations with q-holonomic functions
In this section, we will formulate and prove a few theorems about q-holonomic functions provided by derivation, addition or multiplication of the given q-holonomic functions.
Theorem 3.1. If f (x) is a q-holonomic function of order n, then the function hm (x) =
Dqm f (x) is a q-holonomic function of order at most n for every m ∈ N. Furthermore,
there is an algorithm to compute the corresponding q-differential equation.
Proof. If we prove the statement for m = 1, the final conclusion follows by mathematical induction.
Let h(x) = Dq f (x), where the function f (x) satisfies (6). If p̃0 (x) ≡ 0, then
immediately h(x) is a q-holonomic function of order n − 1.
Hence, let p̃0 (x) 6≡ 0. Then, by Lemma 2.1, we have
Dqn f (x) =
n−1
X
(n)
pk (x)Dqk f (x) ,
k=0
wherefrom
f (x) =
1
(n)
p0 (x)
=
1
(n)
p0 (x)
Dqn f (x) −
n−1
X
(n)
pk (x)Dqk f (x)
k=1
Dqn−1 h(x) −
n−2
X
k=0
7
(n)
pk+1 (x)Dqk h(x) .
Also, by q-deriving, we get
Dqn h(x)
= Dqn+1 f (x) =
n−1
X
(n+1)
pk
(x)Dqk f (x)
k=0
(n+1)
= p0
n−1
X
(x)f (x) +
(n+1)
pk
(x)Dqk−1 h(x)
k=1
=
(n+1)
p0
(x) n−1
Dq h(x)
(n)
p0 (x)
−
n−2
X
n−2
X (n+1)
(n)
pk+1 (x)Dqk h(x) +
pk+1 (x)Dqk h(x).
k=0
k=0
Hence,
Dqn h(x) =
n−1
X
Pk (x; h) Dqk h(x) ,
k=0
where
(n+1)
(n+1)
Pk (x; h) = pk+1 (x) −
(n+1)
Pn−1 (x; h) =
p0
(x)
(n)
p0 (x)
p0
(x)
(n)
p0 (x)
(n)
k = 0, 1, . . . n − 2,
pk+1 (x) ,
.
By multiplying with the common denominator of the rational functions {Pk (x; h), k =
0, 1, . . . , n − 1}, we can conclude that h(x) satisfies the equation
n
X
P̃k (x; h) Dqk h(x) = 0 ,
k=0
i.e it is a q-holonomic function of order ≤ n. ♦
Example 3.1. In Example 2.2, for the q-Pochhammer symbol we proved that it satisfies
f (x) = (x; q)∞
⇒
(1 − x)Dq f (x) +
1
f (x) = 0.
1−q
Now, we have
hm (x) = Dqm (x; q)∞
⇒
(1−q m x)Dq hm (x)+
qm
hm (x) = 0
1−q
(m ∈ N0 ).
Theorem 3.2. If u(x) and v(x) are q-holonomic functions of order n and m respectively, then the functions u(x) + v(x) are q-holonomic functions of order at most
m+n and there is an algorithm to compute the corresponding q-differential equations.
Proof. If u(x) and v(x) are q-holonomic functions of order n and m respectively, they
satisfy holonomic equations
n
X
m
X
p̃k (x) Dqk u(x) = 0 ,
j=0
k=0
8
r̃j (x) Dqj v(x) = 0 ,
(8)
where p̃k (x) and r̃j (x) are polynomials and p̃n 6≡ 0, r̃m 6≡ 0. According to Lemma
2.1, Dql u(x) and Dql v(x) can be represented as
Dql u(x) =
n−1
X
(l)
pk (x) Dqk u(x) ,
Dql v(x) =
m−1
X
(9)
j=0
k=0
(l)
(l)
rj (x) Dqj v(x) ,
(l)
where pk (x) and rj (x) are rational functions as mentioned lemma.
Let h(x) = u(x) + v(x). Then, according to (9), we have
Dql h(x) =
n−1
X
(l)
pk (x) Dqk u(x) +
m−1
X
(l)
rj (x) Dqj v(x) ,
l = 0, 1, . . . , m + n. (10)
j=0
k=0
Taking the values for l = 0, 1, . . . , m + n − 1 in the above identities and expressing
q-derivatives of u(x) and v(x) by q-derivatives of h(x), we get
Dqk u(x)
m+n−1
X
=
(l)
k = 0, 1, . . . , n − 1 ,
(l)
j = 0, 1, . . . , m − 1 .
ak (x) Dql h(x) ,
l=0
Dqj v(x)
m+n−1
X
=
bj (x) Dql h(x) ,
l=0
By eliminating Dqk u(x) (k = 0, 1, . . . , n − 1) and Dqj v(x) (j = 0, 1, . . . , m − 1) from
the last identity (l = m + n) of (10), we get
Dqm+n h(x)
=
m+n−1
X
cl (x)Dql h(x) ,
l=0
where
cl (x) =
n−1
X
(l)
(l)
pk (x)ak (x) +
m−1
X
(l)
(l)
rj (x)bj (x) .
j=0
k=0
By multiplying with the common denominator of {cl (x), l = 0, 1, . . . m + n − 1}, we
get the holonomic equation for h(x)
m+n
X
c̃l (x)Dql h(x) = 0 .
l=0
This proves that the q-holonomic order of u(x)+v(x) is at most m+n, but can be less.
An iterative version of the given algorithm will determine the q-holonomic equation of
lowest order for u(x) + v(x). ♦
Note that the algorithm given in Theorem 3.2 finds a q-differential equation which
is not only valid for u(x)+v(x), but also for every linear combination λ1 u(x)+λ2 v(x),
in particular for u(x) − v(x).
9
Example 3.2. The small q-exponential function from Example 2.3 is q-holonomic of
first order and satisfies
1
u(x)
(1 − q)k
u(x) = eq (x) ⇒ Dqk u(x) =
(k = 0, 1, . . .).
Also, the q-sine from Example 2.5 is q-holonomic of second order and satisfies
v(x) = sinq (x) ⇒ Dqk+2 v(x) =
−1
Dk v(x) (k = 0, 1, . . .).
(1 − q)2 q
Now, by the algorithm given in the proof of Theorem 3.2, the function h(x) = u(x) +
v(x) satisfies
Dq3 h(x) =
1
1
1
Dq h(x) +
h(x).
Dq2 h(x) −
2
1−q
(1 − q)
(1 − q)3
i.e., it is q-holonomic of third order.
Theorem 3.3. If u(x) and v(x) are q-holonomic functions of order n and m respectively, then the function u(x) · v(x) is q-holonomic of order at most m · n and there is
an algorithm to compute the corresponding q-differential equation.
Proof. If u(x) and v(x) are q-holonomic functions of order n and m respectively, they
satisfy holonomic equations (8), and their q-derivatives (9).
Let h(x) = u(x) · v(x). Then, according to (1.5), we have
Dql h(x)
=
l X
l
X
ανµ (x)Dqν u(x)Dqµ v(x)
ν=0 µ=0
=
l X
l
X
n−1
m−1
X (ν)
X (µ)
k
ανµ (x)
pk (x) Dq u(x)
rj (x) Dqj v(x) ,
ν=0 µ=0
i.e.
Dql h(x) =
j=0
k=0
n−1
X m−1
X
(l)
βkj (x)Dqk u(x)Dqj v(x)
(l = 0, 1, . . . , mn),
(11)
k=0 j=0
where
(l)
βkj (x) =
l X
l
X
(ν)
(µ)
ανµ (x)pk (x)rj (x) .
ν=0 µ=0
Taking the relations for (11) for l = 0, 1, . . . , mn − 1 and expressing the q-derivatives
Dqk u(x)Dqj v(x) by q-derivatives of h(x), we get
Dqk u(x)Dqj v(x) =
mn−1
X
(l)
γkj (x) Dql h(x)
l=0
10
(0 ≤ k ≤ n − 1; 0 ≤ j ≤ m − 1).
Eliminating all those Dqk u(x)Dqj v(x) from the last identity (l = mn) of (11), it becomes
mn−1
X
Dqmn h(x) =
σl (x)Dql h(x) ,
l=0
where
n−1
X m−1
X
σl (x) =
(l)
(l)
βkj (x)γkj (x) .
k=0 j=0
By multiplying with the common denominator of {σl (x), l = 0, 1, . . . mn − 1}, we get
the q-holonomic equation for h(x)
mn
X
σ̃l (x)Dql h(x) = 0 .
l=0
This proves that the q-holonomic order of u(x) · v(x) is at most mn, but can be less.
An iterative version of the given algorithm will determine the q-holonomic equation of
lowest order for u(x) · v(x). ♦
Note furthermore that by Lemma 2.2 there is no similar algorithm for the quotient
u(x)/v(x).
Example 3.3. We use again u(x) = eq (x) and v(x) = sinq (x). Now, by the given
algorithm the function h(x) = u(x) · v(x) satisfies
(1 − q)2 Dq2 h(x) − (1 − q 2 )Dq h(x) + qx2 − (1 + q)(x − 1) h(x) = 0,
i.e., it is q-holonomic of second order.
Theorem 3.4. If u(x) is a q-holonomic function of order n, then the function w(x) =
u(xν ) (ν ∈ N) is a q-holonomic function of order at most n and there is an algorithm
to compute the corresponding q-differential equation.
Proof. By assumption u(t) satisfies a q-holonomic equation
n
X
p̃k (t) Dqk u(t) = 0 ,
(12)
k=0
where p̃k (t) are polynomials and p̃n 6≡ 0. Then, by Lemma 2.1, Dql u(t) can be represented as
n−1
X (l)
Dql u(t) =
pk (t) Dqk u(t) ,
(13)
k=0
(l)
where pk (t) are rational functions determined by that lemma.
Let t = xν . Using Lemma 1.3, we have
Dq w(x) = Dqν u(t) Dq (xν ) =
11
u(t) − u(q ν t)
[ν]q xν−1 .
(1 − q ν )t
According to (4), we get
Dq w(x) =
ν
X
ej,ν (x)Dqj u(t) ,
j=1
where
j−1
ej,ν (x) = (−1)
j−1
(1 − q)
j
ν
q (2) xνj−1 ,
j q
j = 1, 2, . . . , ν.
(14)
By (13), we can write
Dq w(x) =
ν
X
ej,ν (x)
j=1
where
(1)
fk,ν (x) =
n−1
X
(j)
pk (t) Dqk u(t) =
k=0
ν
X
n−1
X
(1)
fk,ν (x) Dqk u(t) ,
k=0
(j)
pk (xν )ej,ν (x) ,
k = 0, 1, . . . , n − 1 .
(15)
j=1
Furthermore,
Dq2 w(x)
=
n−1
X
(1)
Dq fk,ν (x)Dqk u(t)
k=0
=
n−1
X
(1)
Dq fk,ν (x) Dqk u(t) +
k=0
n−1
X
(1)
fk,ν (qx)Dq Dqk u(t) ..
k=0
As before, the second sum in the above term can be transformed to
n−1
X
(1)
fi,ν (qx)Dq
Dqi u(t)
=
n−1
X
(1)
fi,ν (qx)
i=0
i=0
=
ν
X
ej,ν (x)Dqj Dqi u(t)
j=1
n−1
ν
XX
(1)
fi,ν (qx)ej,ν (x)Dqi+j u(t)
i=1 j=1
=
ν
n−1
XX
(1)
fi,ν (qx)ej,ν (x)
i=1 j=1
n−1
X
(i+j)
pk
(t) Dqk u(t) .
k=0
Hence,
Dq2 w(x)
=
n−1
X
(2)
fk,ν (x) Dqk u(t) ,
k=0
where
(2)
(1)
fk,ν (x) = Dq fk,ν (x) +
n−1
ν
XX
(1)
(i+j)
fi,ν (qx)ej,ν (x)pk
i=0 j=1
12
(xν ),
k = 0, 1, . . . , n − 1.
By induction, we obtain the representations
Dql w(x) =
n−1
X
(l)
fk,ν (x) Dqk u(t) ,
l = 0, 1, 2, . . . , n
(16)
k=0
(0)
(1)
where fk,ν (x) = δk0 , fk,ν (x) is given in (15) and
(l)
(l−1)
fk,ν (x) = Dq fk,ν (x) +
n−1
ν
XX
(l−1)
fi,ν
(i+j)
(qx)ej,ν (x)pk
(xν ) .
(17)
i=0 j=1
Taking the first n of the identities (16), we can determine
Dqk u(t)
=
n−1
X
(k)
bl,ν (x)Dql w(x) ,
k = 0, 1, . . . , n − 1,
l=0
(k)
where bl,ν (x) are rational functions. Substituting this in identity (16), we get
Dqn w(x) =
n−1
X
(l)
fk,ν (x)
k=0
n−1
X
(k)
bl,ν (x)Dql w(x) =
l=0
where
cl,ν (x) =
n−1
X
cl,ν (x)Dql w(x) ,
l=0
n−1
X
(l)
(k)
fk,ν (x)bl,ν (x).
k=0
By multiplying with the common denominator of {cl,ν (x), l = 0, 1, . . . , n − 1}, we
obtain
n
X
c̃l,ν (x)Dql w(x) = 0 . ♦
l=0
Example 3.4. In Example 2.1, it was proved that
u(x) = xs ⇒ (q − 1) xDq u(x) − (q s − 1) u(x) = 0.
Hence n = 1, p̃1 (x) = (q − 1) x and p̃0 (x) = −(q s − 1). By the procedure of
Theorem 3.4, we get for w(x) = u(x)ν
(1)
Dq w(x) = f0,ν (x)w(x)
where
(1)
f0,ν (x) =
q sν − 1 1
· .
q−1 x
Finally,
w(x) = u(xν ) ⇒ (q − 1) xDq w(x) − (q sν − 1) w(x) = 0.
Example 3.5. In Example 2.2, it was proved that
u(x) = (x; q)λ ⇒ (q − 1) (x − 1)Dq u(x) − (q λ − 1) u(x) = 0.
13
Using our algorithm we get for w(x) = u(x2 ) = (x2 ; q)λ the q-holonomic equation
(q − 1)(x − 1)(x + 1)(x2 q − 1) Dq w(x) − x(q λ − 1)(x2 q λ+1 − q − 1 + x2 q) f (x) = 0
and similar, but more complicated, equations for (xν ; q)λ for higher ν ∈ N.
Example 3.6. In Example 2.5, for the q-sine function, we got
u(x) = sinq (x) ⇒ (1 − q)2 Dq2 u(x) + u(x) = 0.
Now, for w(x) = u(x2 ), we have
(1)
(1)
Dq w(x) = f0,2 (x)u(t) + f1,2 (x)Dq u(t),
with
(1)
f0,2 (x) =
and
qx3
,
1−q
(1)
f1,2 (x) = (1 + q)x
(2)
(2)
Dq2 w(x) = f0,2 (x)u(t) + f1,2 (x)Dq u(t),
with
(qx)2 (−2 − q − q 2 + q 3 x4 )
(1 − q)2
(1 + q)(1 − q + q 2 (1 + q 2 )x4 )
(2)
.
f1,2 (x) =
1−q
By eliminating Dq u(t), we get
(2)
f0,2 (x) =
Dq2 w(x) = c0,2 (x)w(x) + c1,2 (x)Dq w(x),
wherefrom we get for the function w(x) = u(x2 ) the following equation
1 − q4
1 + q2 4 q2
4
xDq2 w(x) − 1 + q 2
x Dq w(x) + qx3
+
x
w(x) = 0.
1−q
(1 − q)3
(1 − q)2
4
Sharpness of the algorithms
In the previous section we proved that the sum, product and composition with powers
of q-holonomic functions are q-holonomic too. In this section we show that the given
bounds for the orders are sharp in all algorithms considered.
Example 4.1. The functions u(x) = x2 and v(x) = x3 are q-holonomic of first order.
According to Theorem 3.2, the function h(x) = u(x) + v(x) is q-holonomic of order
at most two. However, all polynomials are q-holonomic functions of first order, and we
find that h(x) satisfies the equation
x(1 + x)Dq h(x) − ([2]q + [3]q x)h(x) = 0.
This example shows that the order of the sum of some q-holonomic functions can
be strictly less than the sum of their orders. This applies if the two functions u(x) and
v(x) are linearly dependent over K(q)(x).
However, we will prove that for every algorithm given in the previous section there
are functions for which the maximal order is attained.
14
Lemma 4.1. The functions Eq (xµ ) (µ = 1, 2, . . . , n) are linearly independent over
K(q)(x).
Proof. Let us consider a linear combination
r1 Eq (x) + r2 Eq (x2 ) + · · · + rn Eq (xn ) = 0 ,
where rµ = rµ (x) (µ = 1, 2, . . . , n) are rational functions and suppose that rν 6≡ 0.
Then,
n
X
rν Eq (xµ ) = −
rµ Eq (xµ ) ,
µ=0
µ6=ν
i.e.,
n
X
rµ Eq (xµ )
= −1 .
rν Eq (xν )
µ=0
(18)
µ6=ν
Since
A(m) = lim
x→∞
n
m
X
q( 2 )
(xµ )n
(q;
q)
n
n=0
m
X
n
q( 2 )
(q; q)n
n=0
we have
m(µ−ν)
= lim x
x→∞
=
(xν )n
Eq (xµ )
= lim A(m) =
x→∞ Eq (xν )
m→∞
lim
+∞ ,
0,
+∞ ,
0,
µ>ν,
µ<ν,
µ>ν,
µ<ν.
This is a contradiction with (18). Hence, it follows that rµ ≡ 0 for all µ = 1, 2, . . . , n,
i.e. Eq (xµ ) (µ = 1, 2, . . . , n) are linearly independent over K(q)[x]. ♦
Lemma 4.2. The function
Fn (x) =
n
X
Eq (xµ )
(19)
µ=1
is q-holonomic of order n.
Proof. The function Eq (x) satisfies the q-holonomic equation of first order (see Example 2.4)
(1 − q)(t + 1)Dq f (t) − f (t) = 0 .
With respect to Theorem 3.4, for each µ ∈ N, the function Eq (xµ ) is q-holonomic of
first order and one has
(l)
(20)
Dql Eq (xµ ) = f0,µ (x)Eq (xµ ) , l = 0, 1, . . . ,
(l)
where f0,µ (x) are rational functions given as in (17).
15
According to Theorem 3.2, the function Fn (x) is q-holonomic of order at most n.
Therefore
n
n
X
X
(l)
l
l
µ
Dq Fn (x) =
Dq Eq (x ) =
f0,µ (x)Eq (xµ ) .
µ=1
µ=1
Let us suppose that the function Fn (x) satisfies a q-holonomic equation of order m,
i.e.
m−1
X
Dqm Fn (x) +
Ai Dqi Fn (x) = 0 .
(21)
i=0
This equation can be represented in the form
n X
(m)
f0,µ (x) +
µ=1
m−1
X
(i)
Ai f0,µ (x) Eq (xµ ) = 0.
i=0
Since Eq (xµ ) (µ = 1, 2, . . . , n) are linearly independent over K(q)[x], it follows that
(m)
f0,µ (x) +
m−1
X
(i)
Ai f0,µ (x) = 0 ,
µ = 1, 2, . . . , n .
i=0
This can be written in the form of the system of equations
m−1
X
(i)
(m)
Ai f0,µ (x) = −f0,µ (x) ,
µ = 1, 2, . . . , n
i=0
with unknown rational functions Ai = Ai (x).
If m < n, then the system is overdetermined and has no solution. Hence it follows
that m = n. ♦
Theorem 4.3. For each n ∈ N there is a function F which is q-holonomic of order n,
such that H = Dq F is q-holonomic of order n.
Proof. The function defined by (19) satisfies the statement. ♦
Theorem 4.4. For each n, m ∈ N there are functions U and V that are q-holonomic
of order n and m respectively, such that H = U + V is q-holonomic of order n + m.
Proof. Consider the functions
U (x) =
n
X
Eq (xµ )
and
V (x) =
µ=1
n+m
X
Eq (xµ ) .
(22)
µ=n+1
According to Lemma 4.2, they are q-holonomic of order n and m respectively, and the
function
n+m
X
H(x) = U (x) + V (x) =
Eq (xµ )
µ=1
is q-holonomic of order n + m. ♦
16
Theorem 4.5. For each n, m ∈ N there are functions U and V that are q-holonomic
of order n and m respectively, such that H = U · V is q-holonomic of order n · m.
Proof. The statement is valid for the functions defined by (22), because in the function
H(x) = U (x) · V (x) =
n n+m
X
X
Eq (xµ )Eq (xν )
µ=1 ν=n+1
there are nm linearly independent summands Eq (xµ )Eq (xν ) (µ = 1, 2, . . . , n; ν =
n + 1, n + 2, . . . , n + m) over K(q)[x]. The proof of their independence is again based
on Lemma 4.1. ♦
Theorem 4.6. For each n ∈ N there is a function F which is q-holonomic of order n,
such that W (x) = F (xν ) is q-holonomic of order n.
Proof. Starting from the function Fn (x) defined by (19), we can form
W (x) = Fn (xν ) =
n
X
Eq (xµν )
µ=1
which is of the same type as Fn (x). ♦
References
[1] H. B ÖING , W. KOEPF, Algorithms for q-hypergeometric summation in computer
algebra, Journal of Symbolic Computation 28, 1999, 777–799.
[2] G. G ASPER , M. R AHMAN, Basic Hypergeometric Series, 2nd ed, Encyclopedia
of Mathematics and its Applications 96, Cambridge University Press, Cambridge,
2004.
[3] W. H AHN, Lineare Geometrische Differenzengleichungen, 169 Berichte der
Mathematisch-Statistischen Sektion im Forschungszentrum Graz, 1981.
[4] V. K AC , P. C HEUNG, Quantum Calculus, Springer, New York, 2002.
[5] R. KOEKOEK , R.F. S WARTTOUW, Askey-scheme of hypergeometric orthogonal
polynomials and its q-analogue, Report of Delft University of Technology No 9817, 1998.
[6] W. KOEPF, Hypergeometric Summation – An algorithmic approach to Summation and Special Function Identities, Advanced Lectures in Mathematics, Vieweg,
Braunschweig–Wiesbaden, 1998.
[7] W. KOEPF, D. S CHMERSAU, Spaces of Equations Satisfying Simple Differential
Equations, Konrad-Zuse-Zentrum Berlin (ZIB), Technical Report TR 94-2, 1994.
[8] S ALVY, B., Z IMMERMANN , P., GFUN: A package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software 20, 1994, 163–177.
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