Recurrence relations for the connection coecients
of orthogonal polynomials of a discrete variable on the lattice
x
(
s
) =
q
2sStanis law Lewanowicz∗
Institute of Computer Science, University of Wroc law, 51-151 Wroc law, Poland
Received 30 October 1997; received in revised form 30 March 1998
Abstract
We give explicitly recurrence relations satised by the connection coecients between two families of the classical orthogonal polynomials of a discrete variable on a non-uniform latticex(s) =q2s(i. e., theq-analogues of Charlier, Meixner,
Krawtchouk and Hahn polynomials), in terms of the coecients and of the Pearson equation satised by the weight function %, and the coecients of the three-term recurrence relation and of two structure relations obeyed by these polynomials. c1998 Elsevier Science B.V. All rights reserved.
AMS classications:primary 33C45, 33E30
Keywords:Classical orthogonal polynomials of a discrete variable;q-Meixner polynomials;q-Krawtchouk polynomials; Connection coecients; Recurrence relations
1. Introduction
Let{Pk(x)}be any system of the classical orthogonal polynomials of a discrete variable, orthogonal on the exponential lattice x=x(s) :=q2s (s∈ {0;1; : : : ; B−1}) with q= e!,
B−1
X
s=0
Pk(x(s))Pl(x(s))%(s)x(s−1=2) =kld2k (k; l= 0;1; : : :);
where%(s)x(s−1=2)¿0 (s= 0;1; : : : ; B−1), i.e.,q-Charlier polynomialsCk()(x;q); q-Meixner poly-nomials Mk(; )(x;q); q-Krawtchouk polynomials Kk(p)(x;N; q), orq-Hahn polynomials Q(k; )(x;N; q). Here B equals +∞;+∞; N + 1 and N, respectively.
∗E-mail: [email protected].
We are looking for a formula of the type
Pn=
n
X
k=0
cn; kPk; (1.1)
where {Pk} and {Pk} are any two families of classical orthogonal polynomials.
The coecients cn; k in (1.1) are called the connection coecients between the polynomials {Pk} and {Pk} (see [4], Lecture 7).
In a recent paper [3], an algorithmic way has been proposed of obtaining a recurrence relation (in k) of the form
Lcn; k≡ r
X
i=0
Ai(k)cn; k+i= 0: (1.2)
Now, the coecients cn; k can be found by use of this recurrence relation in the backward direction (see [11], Section 7.2).
In the present paper we propose an alternative technique of derivation of the recurrence rela-tion (1.2), based on an generalizarela-tion of an idea introduced in [8] (see also [5]). The dierence operator L is given in terms of the coecients and of the dierence Pearson equation for the
weight %, and the coecients of the three-term recurrence relation and of two structure relations obeyed by {Pk} (see Theorems 3.1 and 3.6). Also, it should be stressed that the order r of the obtained recurrence relation is signicantly lower than in [3]. Applications of the result to some pairs of the classical discrete orthogonal polynomials are given.
2. Properties of the classical orthogonal polynomials
2.1. Basics of classical orthogonal polynomials of a discrete variable
For the sake of compactness, the following notation will be used in the sequel:
D:=
x(s); (2.1)
ˆ
D:=
x(s)ˆ ; x(s) :=ˆ x(s−1=2); (2.2)
N:= B
Bx(s); (2.3)
+(s) :=(s) +(s)x(s);ˆ −(s) :=(s); (2.4)
U:=q−1
−N+I; (2.5)
V:=q+D+I: (2.6)
Here :=E−I; B:=I −E−1; Em (m∈Z) is the mth shift operator, Emf(s) =f(s+m); I is the identity operator, If(s) =f(s). The meaning of and is given below. (By convention, all the
In the sequel, we make use of certain properties enjoyed by all classical families of orthogo-nal polynomials on the lattice x(s) =q2s ([9, Chapter II]; [10]; [1–3, 6]). Besides the three-term recurrence relation
x(s)Pk(x(s)) =0(k)Pk−1(x(s)) +1(k)Pk(x(s)) +2(k)Pk+1(x(s))
(k= 0;1; : : :; P−1(x(s))≡0; P0(x(s))≡1) (2.7)
we need ve other properties.
First, the weight function % satises a dierence equation of the type
ˆ
D[(s)%(s)] =(s)%(s); (2.8)
where (s) := ˜(x(s)); (s) := ˜(x(s)), and where ˜;˜ are polynomials in x; deg ˜62; deg ˜= 1. Second, for arbitrary n, the polynomial Pn obeys the second order dierence equation
LnPn(x(s))≡ {(s) ˆDN +(s)D+nI}Pn(x(s)) = 0: (2.9)
Third, we have the dierence analogue of the Rodrigues formula:
Pk(x(s)) =
Fourth, we have a pair of the so-called structure relations [3],
+(s)DPk(x(s)) =0(k)Pk−1(x(s)) +1(k)Pk(x(s)) +2(k)Pk+1(x(s)); (2.14)
and
Here
2.2. Identities involving the discrete Fourier coecients
We shall need certain properties of the Fourier coecients of an arbitrary polynomial P, degP¡B, dened by
i.e., the coecients in the expansion P=PdegP
k=0 ak[P]Pk.
(cf. (2.7), (2.14) and (2.15), respectively) where I is the identity operator, and Em – the mth shift
operator: Ibk[f] =bk[f]; Embk[f] =bk+m[f] (m∈Z). For the sake of simplicity, we write E in
place of E1. (We adopt the convention that all the script letter operators act on the variable k).
Lemma 2.1. The coecients (2.20) obey the identities:
Here P stands for P(x(s)).
Proof. We shall use the notation
p(s) :=P(x(s)); pk(s) :=Pk(x(s)):
In view of (2.7) and (2.21), identity (2.24) is obviously true.
We will prove the identity (2.25). Using (2.14), summing by parts, and then using (2.18) and the equation
The proof of (2.26) goes as follows:
Similarly, we obtain
Hence follows the identity (2.27).
Identity (2.28) may be proved in an analogous way. Using (2.11), (2.27), and (2.26), we have
bk[LnP] =bk[UDP] +nbk[P] =−qDbk[DP] +nbk[P] = (n−k)bk[P]:
This proves the validity of (2.29).
Remark 1. In the proof of Lemma 2.1 we use the fact that identity (2.24) can be easily generalized
to the form
bk[#P] =#(X)bk[P]; (2.30)
where # is any polynomial in x.
3. Main result
Let {Pk} and {Pk} be any two families of the classical discrete orthogonal polynomials. We shall give a recurrence relation (in k) of the form
Lcn; k≡ r
X
i=0
Ai(k)cn; k+i= 0; (3.1)
obeyed by the connection coecients cn; k in
Obviously, cn; k are the Fourier coecients ak[ Pn]. Let us write
bn; k:=bk[ Pn] =d2kcn; k: (3.3)
Let Pn satises Eq. (2.9), and let
LnPn(x(s))≡ {(s) ˆ DN + (s)D+ nI}Pn(x(s)) = 0: (3.4)
Here (s) :=∗(x(s)); (s) := ∗(x(s)); ∗ and are polynomials in x; deg 62; deg = 1. The constant n is given by n:= −[n]q{12[n−1]q∗′′+ +∗′cosh(n−1)!}, where ∗+(x) :=∗(x) +12(q−
q−1)x∗(x). We shall use the notation
’(x) :=∗(x)−(x);˜ (3.5)
(x) :=∗(x)−(x):˜ (3.6)
We can write
Ln=Ln+’(x(s)) ˆDN + (x(s))D+ ( n−n)I: (3.7)
3.1. Connection between q-Charlier, q-Meixner and q-Krawtchouk families
Now we consider the case where noneof the families {Pk}, {Pk}is a Hahn family. We will prove the following:
Theorem 3.1. Let {Pk}; {Pk} be (independently chosen) families of q-Charlier; or q-Meixner; or q-Krawtchouk polynomials. The coecients (3.3) satisfy the recurrence relation
˜
Lbn; k= 0; (3.8)
where the dierence operator L˜ is given by
˜
L:=D(
kI) +q−1k (q−2X) (3.9) with
k:= n−k− ′q−2: (3.10)
The order of the recurrence relation (3.8) is not greater than 2.
Proof. Under the assumptions of the theorem, we have ∗= ˜ (cf., e.g., [9, Table 3.3], or [1]), so that Eq. (3.7) simplies to
LnPn(x(s)) =LnPn(x(s)) + (x(s))DPn(x(s)) + ( n−n) Pn(x(s));
which can be rewritten in the form
LnPn(x(s)) =LnPn(x(s)) +D[ (x(s−1)) Pn(x(s))] +–Pn(x(s))
with –:= n−n−D (x(s−1)) = n−n− ′q−2. Using this result in the equation
we obtain
bk[LnPn] +bk[D{ (q−2x) Pn}] +bk[–Pn] = 0:
Applying the operator D to both sides of the above equation, and making a repeated use of
Lemma 2.1, we arrive at the recurrence relation (3.8) with the operator ˜L given in (3.9).
Corollary 3.2. The connection coecients in (1.1) satisfy
Lcn; k= 0 (3.12)
with
L:=d−2 k L˜(d
2
kI); (3.13)
where L˜ is the dierence operator given in (3.9).
Remark 2. Notice that in (3.13) we need not the explicit form for d2
k, but only for the quotients d2
k+h=d2k. Using the equation d2k+1=d2k=0(k+ 1)=2(k) (see [9, p. 106]), we obtain
d2 k+h d2
k =
h
Y
m=1
0(k+h)
2(k+h−1)
(h¿0):
In particular, for the monic case we have 2≡1. Making use of the forms of the operators D
(see (2.22)) and X (see (2.21)), we arrive at the following:
Corollary 3.3. Scalar form of Eq. (3.12) is
A0(k)cn; k−1+A1(k)cn; k+A2(k)cn; k+1= 0; (3.14)
where
A0(k) :=0(k)( n− ′−k−1) +q−3k ′0(k);
A1(k) :=0(k){1(k)( n− ′−k) +q−1k[ ′q−21(k) + (0)]};
A2(k) :=0(k)0(k + 1){2(k)( n− ′−k+1) +q−3k ′2(k)}:
(3.15)
Example 3.4. Let us consider the following formula, connecting two q-Meixner families:
Mn;(x;q) = n
X
k=0
cn; kMk; (x;q): (3.16)
The specic expressions for ; ; k as well as the forms for the coecients of the operators X (see (2.21)) and D (see (2.22)) for the monic q-Meixner polynomials are given in Table 1 (see
The coecients cn; k in (3.16) satisfy Eq. (3.14) with Ai’s given in (3.15), where, in particular,
n:=−[n]qq$[n+ $−1]q;
′:=q$[ $]
q−q$[$]q; (0) :=q+1[+ 1]q−q
+1[ + 1] q; q:= ; $ := + + 1:
Example 3.5. Let us consider the following formula, connecting two q-Charlier families:
C( )
n (x;q) = n
X
k=0
cn; kC ()
k (x;q): (3.17)
The specic expressions for ; ; k as well as the forms for the coecients of the operators X (see (2.21)) and D (see (2.22)), and N (see (2.23)) for the q-Charlier polynomials are given in
Table 2 (see Appendix A). The coecientscn; k in (3.17) satisfy Eq. (3.14) withAi’s given in (3.15), where, in particular,
n=n:=q1−n[n]q=–q; ′:= 0; (0) := ( −)q3:
Here –q:=q−q−1. Noticing that the coecient
2 of the operator D vanishes we see that A2≡0,
so that we obtain the rst-order recurrence
A0(k)cn; k−1+A1(k)cn; k= 0;
where
A0(k) = ( n−k−1)0(k);
A1(k) =0(k){( n−k)1(k) +q2( −)k}:
Hence the formula
cn; k= (−–q)n−k
n n Y
j=k+1
q2j−1( n−j)1(j) +q2( −)j
n−j−1
(06k6n):
3.2. Case when a q-Hahn family is involved
Observe that for arbitrary pair of classical orthogonal sequences we have
˜
(x) =(x)(x−1); ∗(x) =(x)(x−1); (3.18)
The case 6= is discussed in the following theorem.
Theorem 3.6. Let {Pk} and {Pk} be such two sequences of classical orthogonal polynomials that (i) exactly one of the sequences belongs to the Hahn family; or (ii) {Pk}; {Pk} are both in the Hahn family; Pk=Qk(·;; ; N; q); Pk=Qk(·; ;; N ; q); with 6= ; or N6= N.
The coecients bn; k:=bk[ Pn] =d2kcn; k; where cn; k’s are the connection coecients in
Pn=
n
X
k=0
cn; kPk; (3.19)
satisfy the fourth-order recurrence relation
˜
Lbn; k= 0; (3.20)
where the dierence operator L˜ is given by
˜
L:=D(X)(n−k)I+q−1
k#(q−2X) +D(X); (3.21)
with
#:=∗−;˜ :=
n−n; (x) :=(x)−D#(q−2x): (3.22)
Proof. Notice that the assumptions of the theorem imply that6= (cf., e.g., [9, Table 3.3], or [1]). Multiplying both sides of (3.4) by , and making use of ∗=, we write˜
LnPn(x(s)) =LnPn(x(s)) +#(x(s))DPn(x(s)) +Pn(x(s))
=LnPn(x(s)) +D[#(x(s−1)) Pn(x(s))] +Pn(x(s));
notation used being that of (3.22). Now, in view of (3.4) we have
0 =bk[LnPn] +bk[D{#(q−2x) Pn}] +bk[Pn]:
Applying the operator D to both sides of the above equation, and making a repeated use of
Lemma 2.1, we arrive at the recurrence relation (3.20) with the operator ˜L given in (3.21).
Acknowledgements
The author wishes to thank the referees for their valuable comments.
Appendix A
Table 1
Data for the monicq-Meixner polynomials [1, 2, 6] Mk; (x;q); x=q2s; =q2
Data for theq-Charlier polynomials [1, 2]
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