Let us consider the following inner product defined in the linear space !#&#34

CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF ORDER (II)

M. ISABEL BUENO, FRANCISCO MARCELLÁN,ANDJORGE SÁNCHEZ-RUIZ

Abstract. Given a symmetrized Sobolev inner product of order , the corresponding sequence of monic or- thogonal polynomials satisfies , for certain sequences of monic polynomials and . In this paper we consider the particular case when all the measures that define the symmetrized Sobolev inner product are equal, absolutely continuous and semiclassical. Under such restrictions, we give explicit algebraic relations between the sequences and , as well as higher-order recurrence relations that they satisfy.

Key words. Sobolev inner product, orthogonal polynomials, semiclassical linear functionals, recurrence rela- tion, symmetrization process

AMS subject classification.42C05

1. Introduction. Let us consider the following inner product defined in the linear space

!#"$!

, where^! denotes the linear space of polynomials with real coefficients, (1.1) %'&)(+*-,/.10

23

&4*65879;:=<

>?@BA6C

?

23

&)D

?E

* D

?E

587

?GF

In the previous expression ^{7H9 (I7}

A ( FFF

(I7

<

denote positive and absolutely continuous Borel measures supported on a subset of the real line and such that the corresponding sequences of moments are finite,^{& D}

?E

denotes the^Jth derivative of^& , and^C

?

are nonnegative real numbers (^C

<LK

0NM ). The inner product given in (1.1) is known asSobolev inner product of order [7]. Sobolev inner products and their corresponding sequences of orthogonal polynomials have been exhaustively studied during the last ten years, although most of the results have been obtained for ^0PO .

The product ^{%IQ(RQ, .} is said to besymmetrizedif ^{%TSU(ISVW, . 0XM} when^YZ:\[ is an odd nonnegative integer. In this case,^{7 9} ,⁷

A

,...,⁷

<

are supported on a subset of the real line which is symmetric with respect to the origin and the measures themselves are also symmetric, so

that ^]

D

?E

^

U-_

A 0 2 3 S ^

U-_

A

587

?

0`Ma(bJc0`Md(eO8(

FFF

( (fYhgi

F

This concept extends the definition ofsymmetric linear functional[5] to the bilinear case.

Assume that^%IQ(RQ,/. is quasi-definite, that is, there exists a sequence ^jlk

Um

of polynomi- als orthogonal with respect to ^%nQ(eQo,+. . If^%nQ(eQo,/. is symmetrized, then there exist another two sequences of polynomials^jp

Um

and^jq

Um

such that

(1.2) ^{k ^}

Ur Ss0`p

Ur S ^

st(uk

^

U-_

A r

Ss0`Svq

UHr S ^ s F

In the sequel, we will refer to these two sequences as thesymmetric components of^jwk

U m

. In this paper we consider the bilinear symmetrization problem associated with (1.1), i.e.

the analog in the bilinear case of Chihara’s linear symmetrization problem [5], which consists in:^x

Received December 15, 2004. Accepted for publication Octuber 31, 2005. Recommended by R. Alvarez- Nodarse.

Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain ([email protected], [email protected], [email protected]).

Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain.

55

(1) finding the explicit expressions of the bilinear functionals such that^jp

Um

and^jq are the corresponding sequences of orthogonal polynomials, Uvm

(2) determining recurrence relations with a finite number of terms that^jlp

U m

and^jq

U m

satisfy, and

(3) obtaining explicit algebraic relations between both sequences.

The problem (1) was solved for Sobolev inner products of order^O in [3], and for general in [4]. As regards problems (2) and (3), till now they have only been solved in the particular case when ^0XO and the two measures that define the product^{%nQ(eQo, .} are equal, absolutely continuous and semiclassical [3]. In this paper we extend these results for an arbitrary^{Ly O} , under the same restrictions on the measures involved.

The structure of the paper is the following: First, in Section2, we present some auxiliary results related with semiclassical functionals and semiclassical measures. In Section3, we find explicit algebraic relations between the sequences ^jp

Um

and ^jq

Um

. In Section4, we determine recurrence relations with a finite number of terms that the sequences^jlp

Um

,^jq and^jlk Uvm

Udm

satisfy. Finally, in Section5, we apply our general results to a particular case of the so-called Freud-Sobolev polynomials [2].

2. Auxiliary results. Consider a quasi-definite linear functional^z in^! , with integral representation

(2.1) ^z

r

&s0 2 3 & r

SsI57{0 2 3 & r

Ss}|

r

SsI5Sh(

where⁷ is an absolutely continuous positive Borel measure and^| is the corresponding weight function. The functional^z is said to be asemiclassical linear functionalif

(2.2) ^~

r€

U^s0‚ U⁽ where



and^ are polynomials with^ƒ„R

r€

sW0‡†

y M and^ƒ„e

r

1s

y O , and^~ denotes the derivative operator. The condition of being semiclassical can also be characterized in terms of the weight function^| :

PROPOSITION 2.1. [6] Let ^z be a semiclassical linear functional with integral repre- sentation (2.1), where^| is a continuously differentiable function in an interval^ˆ‰(+Š‹ satisfying

ŒŽl‘“’”•

6r Ss–&

r

Ss}|

r

Ss;0—M with^{&Zg !} . Then,

(2.3)

r˜

|™sIš0`|(

and^| is said to be asemiclassical weight function. Equation (2.3) is the so-called Pearson equation.

DEFINITION2.2.[8] Given a semiclassical linear functional^z , let^›œ be the set of all the pairs of polynomials satisfying (2.2). Then, the class^Ÿž of^z is defined as

(2.4) ^{Ÿž 0 Ž ¡}

D¢

”£ E}¤-¥H¦¨§

Ž©wª

jƒ«„e…

r€

sG¬#­®(/ƒ„R…

r

1sG¬¯O

m±°

F

LEMMA 2.3. [7] If^| is a semiclassical weight function, then for every nonnegative integer ,

 < r Ssn~

<²

| r

SsI³W0\

r

S)( s}|

r

Ss(

where

 r

S)(+M8s0´O(

 S)( s0 Ssn™š SB( ¬¯OsH:µ S)( ¬¯Oseˆ SsG¬ š Ss}‹ (

¶y

O F

L^EMMA 2.4. [7] Given a semiclassical weight function^| , the polynomials^

r

S)( s

defined in the previous lemma satisfy

ƒ«„e…

² r

S)( sI³·

r ž

Ÿ

:`Os(

Ly

Ma(

where^Ÿž is the class of the semiclassical linear functional defined by^| . Consider the following linear differential operator in the linear space^! :

(2.5) ^¸ D

<

E™¹

0 <

>

V @ 9 r

¼Ols

V C V V

>?@

9¨»

[

JH¼



<¾½ V1_

? r

Ss¿

r

S)(/[X¬ÀJnsn~

V1_

? (

where^{C 9 0´O} and^{~ 9 0`Á} , the identity operator.

PROPOSITION2.5.Let^z be a semiclassical linear functional with integral representa- tion (2.1). Using the notations introduced above, for all nonnegative integers and^Y ,

(1) If^{Ÿž ¬À† Â¯M} , then^ƒ„R

²¸ D< E r

SvUds/³·ÃYÄ: †®:

r ž

Ÿ

¬À†ws

Ž ¡

j (IY

m

. (2) If^{Ÿž ¬À† ·¯M} , then^ƒ„R

²¸ D< E r

SvUds ³ 0\YÄ: † .

Proof. From (2.5),

¸ D< E rS U s0 <

>

V @ 9 r

¼OsV C V V

>?@

9¨»

[

J¼



<¾½ V1_

? r

Ssn

r

S)(I[ŬÀJ¿sI~ V1_

? rS U s F

Notice that ^~ÆV1_

?r

SvUdsh0ÇM when^{[È:ÉJh¶Y} . It follows that the upper bounds on the

summations over ^[ and ^J can be replaced, respectively, by ^{Ê 0 Ž¨¡ j (/Y}

m

and Ê [Ë0

Ž¨¡

j[h(/Y{µ[

m

, with the condition that empty sum equals zero if^Y—ÌÍ[ . Using Lemma 2.4,

ƒ„R…

²

¸ D

< E rS U s ³ ·

Ž©wª

?@

9tÎÐÏ

V

V @

9ÐÎ

Ï

<

jR†

r

¬À[Ñ:µJnsH:

r

[Ò¬ÀJ¿s

r ž

Ÿ

:\OlsH:ÓY¬À[Ò¬ÀJ

m

0

Ž©wª

?@

9tÎÐÏ

V

V @

9ÐÎ

Ï

<

jRYÄ: †®:µ[

r ž

Ÿ

¬À†-sc¬ÀJ

r ž

Ÿ

:íW¬h†-s

m F

From (2.4), we know that^{Ÿž :¯­4¬À† y M} . Therefore, the maximum is attained for^J60—M ,

ƒ«„e…

²¸ D< E rS U s ³ ·

Ž¨©-ª

V @

9tÎ

Ï

<

jY: †a:Ó[

r ž

Ÿ

¬h†-s

m F

If^{Ÿž ¬À†¨Â‚M} , then the maximum is attained for^{[Ô0 Ê} , and the result in (1) follows. On the

other hand, if^{Ÿž ¬h†Õ·‚M} , then the maximum is attained for^{[Å0`M</sup> ; since the term<sup>[Å0`JG0\M} in^{¸ D}

< E r

SUds has exact degree^{YÄ: †} , (2) holds.

COROLLARY2.6.Let^Ÿ

¹0

Ž©wª

jR†w(wž

Ÿ m

. Then, for all nonnegative integers ^(/Y ,

ƒ„e…

²¸ D< E rS U s ³

·ÃYÄ:

Ÿ F

3. Explicit algebraic relations between^jp

U m

and^jq

U m

. Let us consider a symmetrized quasi-definite Sobolev inner product given by (1.1) with⁷

?

0‚7 for all^J (^M ·ÃJ· ), i.e., (3.1) %Ö&)(/*-, . 0 2 3

&4*G57\:

<

>?@BA

C ? 2 3

&HD

?E

* D

?E

587

F

We denote by^jwk

Um

the sequence of monic polynomials orthogonal with respect to (3.1), and by^jlp

Um

and^jlq

Um

the corresponding symmetric components defined by (1.2). The weight function^| satisfying^57{0‚|

r

SsI5S and the corresponding linear functional^z given by (2.1)

are both symmetric. Furthermore, the sequence^jR×

U m

of monic polynomials orthogonal with respect to^z satisfies a three-term recurrence relation,

(3.2) ^Sv×

Ur Ss0\×

U-_

A r

SsB:

] U × U ½ A r

Ss(fY

y

O(

where

] U K

0\M , and there exists a sequence of monic polynomials^jlØ

U m

such that

(3.3) ^{× ^}

Ur Ss™0—Ø

Ur S ^

s(f×

^

U_

A r

Ss0`SØ

U rS ^ s(

where^jwØ

U m

denotes the sequence of monic kernel polynomials associated with^jlØ

U m

[5].

In the sequel, we assume that ^z is a semiclassical linear functional, and we use the notations introduced in the previous section. The following proposition states an algebraic relation between the sequences ^jwk

U m

and ^j×

U m

that will be useful to determine algebraic relations between^jp

U m

and^jlq

U m

.

LEMMA3.1.[7] Let^& and^* be arbitrary polynomials. Then,

% <

&B(/*-,/.;0—z

²

&¸ D< E r

*-sI³

F

P^ROPOSITION 3.2. For every nonnegative integer^{Y yÅ Ÿ} , there exist real numbers

Ù U ”Ú

such that

(3.4)

 < r Ss¿×

UHr Ss0

U-_

<1Û

>

Ú@ U

½<

. Ù U ”Ú k Ú r

Ss

F

Proof. Expanding the polynomial

 < r Ss¿×

Ur

Ss in terms of the Sobolev polynomials we get

 < r Ss¿×

U r Ss0

U-_

<1Û

>Ú @ 9 Ù U ”Ú k Ú r

Ss

F

Then, we use Lemma3.1to compute the coefficients^Ù

U ”Ú

,

Ù U ”Ú 0 % < r

Ss¿×

UHr Sst(+k

Ú r

Ss/,.

%˜k Ú r

Sst(+k Ú r

Ss/,. 0 z ²×

Ur Ssn¸

D< E ²k Ú r

Ss³³

%˜k Ú r

Sst(+k Ú r

Ss/, . F

Since^j×

Um

is the sequence of polynomials orthogonal with respect to^z , Corollary2.6im- plies that^Ù

U ”Ú

0\M ifM¨·#ܐ̯Y$¬

Ÿ .

REMARK3.3. If in the above proof Proposition2.5is used instead of Corollary2.6, a sharper lower bound can be obtained for the summation in (3.4). This means that the first terms of the sum in (3.4) may be zero. However, since the use of Proposition2.5would require two parallel developments in what follows, for the sake of brevity we use the general bound given by Corollary2.6.

3.1. Semiclassical functional^z of even class.

P^ROPOSITION3.4.If the class^Ÿž of the functional^z is even, then the following explicit algebraic relation between the sequences^jp

U m

and^jq

U m

is obtained,

V;_

< Ï

Û

>

Ú @ V

½<

Ï

. Ù ^ V ”^ Ú p Ú r

Ss0 Ù ^

V1_

A ”^

V1_

<1Û _ A q

V1_

< Ï Û r

Ss

:

V1_

< Ï Ûн

A

>

Ú @ V

½<

Ï

. rÙ ^ V1_

A ”^Ú _ A : ] ^ V Ù ^ V ½ A ”^ Ú _ A

snq Ú r

Ss

: ] ^ V Ù ^ V ½ A ”^ V

½<

. ½ A q V

½<

Ï.

½ A r Sst(

(3.5)

where Ê

†“0`†-Ý­ and Ê

Ÿ 0 Ÿ

Ý-­ .

Proof. Since^Ÿž is an even number,



is an even polynomial [3, Prop. 2.6]. That is, there exists another polynomial ^ž



such that (3.6)

6r Ss™0

ž

6r S ^ s F

Therefore,^† and^Ÿ are also even numbers. We write^†“0—­ Ê

† and^{Ÿ 0`­} Ê

Ÿ .

For^YZ0—­w[ , Proposition3.2reads

(3.7)

 < r Ss¿×

^ V r

Ss0

^

V;_

<;Û

>

Ú @ ^ V

½<

. Ù ^ V ”Ú k Ú r

Ss

F

Since the term in the left-hand side of the previous identity is an even polynomial, we get

 < r Ss¿×

^

V r Ss™0

V1_

< Ï

Û

>

Ú @ V

½<

Ï.

Ù ^ V ”^Ú k ^Ú r

Ss

F

Taking into account (1.2), (3.3), and (3.6), the previous equation can be rewritten as

(3.8) ^ž

 < r Ss/Ø

V¨r Ss™0

V1_

< Ï

Û

>

Ú @ V

½<

Ï.

Ù ^ V ”^Ú p Ú r

Ss

F

In a similar way, forYÞ0`­w[N:\O we obtain

(3.9) ^ž

 < r Ss/Ø

V r

Ss0 V1_

< Ï

Û

>

Ú @ V

½<

Ï.

Ù ^

V1_

A ”^Ú _ A q Ú r

Ss

F

Taking into account (3.3) and (3.2) with^Y replaced by^­w[ , we get

(3.10) ^Ø

V r Ss0ßØ

V r

Ss):

] ^ V Ø

V ½ A r

Ss

F

Multiplying (3.10) by ^ž

 <

and using (3.8) and (3.9), we obtain

V1_

< Ï

Û

>

Ú @ V

½<

Ï

. Ù ^ V ”^ Ú p Ú r

Ss

0

V1_

< Ï

Û

>

Ú @ V

½<

Ï.

Ù ^

V1_

A ”^Ú _ A q Ú r

Ss:

] ^ V

V1_

< Ï Ûн

A

>

Ú @ V

½<

Ï.

½ A Ù ^ V ½ A ”^ Ú _ A q Ú r

Ss F(3.11)

The result in (3.5) can be obtained in a straightforward way from the previous expression.

3.2. Semiclassical functional^z of odd class.

P^ROPOSITION3.5.Let the class^Ÿž of the functional^z be an odd number. The following explicit algebraic relations between the sequences^jp

U m

and^jlq

U m

are obtained:

(1) If is an even number and^{×\0 Ý­} , then

V1_Hà

Û

>

Ú @ V ½ à . Ù ^ V ”^ Ú p Ú r

Ss0 Ù ^

V1_

A ”^

V1_

<1Û _ A q

V;_)à

Û r Ss

:

V1_Hà

Ût½

A

>

Ú @ V ½ à . rÙ ^

V1_

A ”^Ú _ A : ] ^ V Ù ^ V ½ A ”^Ú _ A

sIq Ú r

Ss

: ] ^ V Ù ^ V ½ A ”^ V

½<

. ½ A q V ½ à . ½ A r

Ss

F

(2) If is an odd number, and^×\0

r

¬¯Os/Ý­ , then

V1_Hà

Û _ Ï

Û _ A

>

Ú @ V ½ à . ½

Ï.

Ù ^

V;_

A ”^ Ú p Ú r

Ss0 Ù ^

V1_

^ ”^

V1_

<1Û _ ^ q

V1_Hà

Û _ Ï

Û _ A r

Ss

:

V;_)à

Û _ Ï

Û

>

Ú @ V ½ à . ½

Ï.

rÙ ^ V1_

^ ”^Ú _ A : ] ^

V1_

A Ù ^ V ”^ Ú _ A

sIq Ú r

Ss

: ] ^

V1_

A Ù ^ V ”^ V

½<

.q

V ½ à . ½

Ï.

½ A r Sst(

where Ê

†0

r

†¬¯Os/Ý­ andÊ

Ÿ 0 rŸ

¬ÃOls/Ý­ .

Proof. Since^Ÿž is an odd number,



is an odd polynomial [3, Prop. 2.6]. Therefore, there exists another polynomial ^Ê



such that (3.12)

6r

Ss0\S Ê

Gr S ^ s F

Since^† and^Ÿž are odd numbers, so it is^Ÿ . In the sequel, we write^†º0ß­ Ê

†;:‚O and^{Ÿ 0ß­} Ê

Ÿ

:\O .

(1) Assume that is even. Then, we write ^0P­-× , for some nonnegative integer^× . From (3.4), we get

 < r Ss¿×

^ V r

Ss0 V1_Hà

Û

>

Ú @ V ½ à . Ù ^ V ”^Ú k ^Ú r

Ss

F

Taking into account (1.2), (3.12), and (3.3), we get

(3.13) ^{S à Ê}

 < r Ss/Ø

V¨r Ss0

V;_)à

Û

>

Ú @ V ½ à . Ù ^ V ”^ Ú p Ú r

Ss

F

Consider Proposition3.2withYÞ0—­-[Í:\O to obtain, in a similar way,

(3.14) ^{S à Ê}

 < r SsIØ

V r

Ss0 V1_Hà

Û

>

Ú@ V ½ à . Ù ^

V1_

A ”^ Ú _ A q Ú r

Ss

F

By multiplying both sides of (3.10) by^SàÊ

 < r

Ss, (3.13) and (3.14) lead us to the following explicit algebraic relation between^jlp

Um

and^jlq

Um

,

V1_Hà

Û

>

Ú @ V ½ à . Ù ^ V ”^ Ú p Ú r

Ss

0

V;_)à

Û

>

Ú @ V à . Ù ^

V1_

A ”^Ú _ A q Ú r

SsB:

] ^ V

V;_)à

Ûe½

A

>

Ú @ V à . A Ù ^ V ½ A ”^ Ú _ A q Ú r

Sst(

(3.15)

and the result follows in a straightforward way.

(2) Assume that is odd. Then ^{0P­-×á:ßO} for some nonnegative integer^× . In this case, the term on the left-hand side of (3.7) is an odd polynomial, and taking into account (1.2), (3.12), and (3.3), we get

(3.16) ^{S à Ê}

 < r SsIØ

V r Ss0

V;_)à

Û _ Ï

Û

>

Ú @ V ½ à . ½

Ï.

½ A Ù ^ V ”^Ú _ A q Ú r

Ss

F

A similar procedure gives, using Proposition3.2withYZ0ß­w[Ñ:`O ,

(3.17) ^{S à_}

A Ê

 < r

SsIØ

V r

Ss0 V;_)à

Û _ Ï

Û_ A

>

Ú @ V ½ à . ½ Ï. Ù ^

V1_

A ”^Ú p Ú r

Ss

F

From (3.2) withYÞ0`­-[Ñ:`O , and (3.3),

(3.18) ^SØ

V r

Ss0ßØ

V1_

A r

Ss6:

] ^

V1_

A Ø V r

Ss

F

Replacing (3.16) and (3.17) into (3.18), the following relation is obtained,

V1_Hà

Û _ Ï

Û_ A

>

Ú@ V ½ à . ½ Ï. Ù ^

V1_

A ”^Ú p Ú r

Ss

0

V;_)à

Û _ Ï Û _ A

>

Ú @ V ½ à . ½

Ï.

Ù ^

V1_

^ ”^ Ú_ A q Ú r

Ss):

] ^

V1_

A

V1_Hà

Û _ Ï

Û

>

Ú @ V ½ à . ½ Ï. ½ A Ù ^ V ”^ Ú _ A q Ú r

SsÐ(

(3.19)

and the result follows straightforwardly.

4. Recurrence relations. In this section we deduce recurrence relations with a finite number of terms for the sequences ^jp

U m

,^jq

U m

, and^jwk

U m

. The number of terms in these relations depends on the class of the functional^z and the degree of the polynomial



. 4.1. Recurrence relations for^jp

Um

PROPOSITION 4.1. If the class ^Ÿž of. ^z is an even number, then the sequence ^jlp satisfies the following [ Um

r Ê

Ÿ : Ê

†sH:Ãâ ]-term recurrence relation,

ãä€å)æ8änçä€å)æ8èé¿æ8äIê å)æ8èëé}æ±ì

D E

@ D

ã8ä˜åcçä€å)æ èé

½

ãä€å)æ8änçä€å)æ èé

½ºí

ä˜å)æ±ìnã8ä€åGçä˜å)æ èé

½í

ä€åãä€åGçä˜åBæ8èé

Eê

åBæ8ècëé

D E

_

å)æ8èëé/î ì

ï

ð˜ñ åGî8ècë

ò

æ±ì¿ó



ã8ä˜åGçäð ½

ã8ä€å)æäIçäð

½í

ä€å)æ±ìnã8ä˜åGçäð

½ºí ä˜å

D

ã8ä€åGçäð _ í

ä˜åGî ì/ã8ä˜åGîäIçäð

Eô

ê ð D E

_ ó

ã8ä˜åcçä€åGî8è

ò

½í

ä€å)æ ìIãä€åGçä€åGî8è

ò

½ºí ä˜å

D

ã8ä€åGçä˜åGî8è

ò _ í

ä€åGî ìnã8ä˜åcîänçä€åGî8è

ò

Eô

ê åGî8èGë

ò D E

½í ä€å

í

ä˜åGî ì

ã

ä˜åGîäIçä˜åcî8è

ò

îä

ê åGî8ècë

ò

î ì

D E”

whereÊ

†º0`†Ý-­ andÊ

Ÿ 0 Ÿ

Ý-­ .

Proof. Assume that^Ÿž is an even number. Multiplying both sides of (3.18) by ^ž

 < r Ss, and plugging (3.8) and (3.9) into it, we get

S`õö V1_

< Ï

Û

>

Ú @ V

½<

Ï.

Ù ^

V1_

A ”^Ú _ A q Ú r

Ss€÷ø

0

V;_

< Ï Û _ A

>

Ú @ V

Ï.

_ A Ù ^

V1_

^ ”^Ú p Ú r

SsÞ:

] ^

V1_

A

V1_

< Ï

Û

>

Ú@ V Ï. Ù ^ V ”^ Ú p Ú r

Ss F(4.1)

Now, we multiply both sides of (3.11) by^S and replace (4.1) in it to obtain

S õö

V1_

< Ï

Û

>

Ú@ V

½<

Ï

. Ù ^ V ”^ Ú p Ú r

Ss÷ø

0

V1_

< Ï Û_ A

>

Ú@ V

½<

Ï.

_ A Ù ^

V1_

^ ”^ Ú p Ú r

SsÞ:

] ^

V;_

A

V1_

< Ï

Û

>

Ú @ V

½<

Ï.

Ù ^ V ”^Ú p Ú r

Ss

: ] ^ V õö

V1_

< Ï

Û

>

Ú @ V

½<

Ï.

Ù ^ V ”^Ú p Ú r

SsB:

] ^ V ½ A

V1_

< Ï Ûe½

A

>

Ú @ V

½<

Ï.

½ A Ù ^ V ½ ^ ”^ Ú p Ú r

Ss

÷ø F

From the previous expression we get the ( Ê

Ÿ : Ê

†¾:Ãâ )-term recurrence relation for^jlp

U m

given in the statement of the proposition.

P^ROPOSITION4.2. Let the class^Ÿž of^z be an odd number, and put Ê

†0

r

†º¬\Ols/Ý­ and

Ê

Ÿ 0 rŸ

¬ùOs+Ý-­ . Then the sequence ^jp

Um

satisfies the following ^ˆ

r Ê

Ÿ : Ê

†º:‡Os:ßâ-‹-term

recurrence relations:

(1) If is an even number, and^{×\0 Ý-­} , then

Ù ^

V1_

^ ”^

V1_

<1Û _ ^ p

V;_)à

Û _ A r

Ss

0‡ˆ

Ù ^ V ”^

V1_

<1Û

r

SĬ

] ^

V1_

A ¬ ] ^ V

sc¬

Ù ^

V1_

^ ”^

V;_

<;Û

‹p

V1_Hà

Û r Ss

:

V1_Hà

Ût½

A

>

Ú @ V ½ à ._ A ˆr

SƬ

] ^

V1_

A ¬ ] ^ V sÙ ^ V ”^Ú ¬ Ù ^

V;_

^ ”^ Ú ¬ ] ^ V ] ^ V ½ A Ù ^ V ½ ^ ”^Ú

‹p Ú r

Ss

:—ˆ

r

SƬ

] ^

V1_

A ¬ ] ^ V s Ù ^V ”^ V

½<

.™¬

] ^ V ] ^ V ½ A Ù ^ V ½ ^ ”^ V

½<

./‹p

V ½ à . r

Ss

¬ ] ^ V ] ^ V ½ A Ù ^ V ½ ^ ”^ V

½<

. ½ ^ p V ½ à . ½ A r

Ss

F

(2) If is an odd number, and^×\0

r

¬¯Os/Ý­ , then

ã ä˜å)æ8úIçä€å)æ èé¿æúêå)æ ûwé¿æëénæ8ä

@ D ã

ä€å)æ±ì}çä€å)æ8èé¿æ±ì

½ ã

ä€å)æúIçä˜åBæ8èé¿æ±ì

½í ä€å)æ8ä

ã

ä˜åBæ ì¿çä˜å)æ èé¿æ ì

½í ä€å)æ±ì

ã

ä˜å)æ±ì¿çä€å)æ èé¿æ ì

Eê å)æ ûwé¿æëénæ ì

D E

_

åBæ8ûé}æëé

ï

ð˜ñ åGî8û

ò

îë

ò

æ ì¿ó

 ã ä€å)æ ì¿çäð ½ ã ä˜å)æ8úIçäð ½ Dí

ä€å)æä

_ í

ä€å)æ ì

Eã ä˜å)æ±ì¿çäð

½ºí ä˜åBæ ì

í

ä˜å ã ä˜åcî ì}çäðôê ð D E

½ D

ã8ä˜åBæ ì¿çä˜åGî8è

ò

æ±ì

½í

ä€å)æ8ä/ã8ä˜å)æ±ì¿çä€åGî8è

ò

æ ì

½í

ä€å)æ±ìnã8ä˜å)æ±ì¿çä€åGî8è

ò

æ ì

½í ä€å)æ±ì

í

ä€åãä€åGî ì¿çä€åGî8è

ò

æ ì

Eê åcî8û

ò

îvë

ò D E

½í ä€å)æ±ì

í

ä€åãä€åGî ì¿çä˜åGî8è

ò

î ì¿ê åGî8û

ò

îë

ò

î ì

D E}ü

Proof. Again, we must distinguish between being odd or even.

(1) Assume that is even. Multiplying both sides of (3.18) by^{Sà Ê}

 < r

Ss and applying (3.13) and (3.14) into it, we get

S`õö V1_Hà

Û

>

Ú @ V ½ à . Ù ^

V1_

A ”^Ú _ A q Ú r

Ss}÷ø

0

V;_)à

Û _ A

>

Ú @ V ½ à ._ A Ù ^

V;_

^ ”^ Ú p Ú r

SsB:

] ^

V1_

A

V1_Hà

Û

>

Ú@ V ½ à . Ù ^ V ”^ Ú p Ú r

Ss F(4.2)

Now, multiplying both sides of (3.15) by^S and using (4.2) in the resulting equation, we obtain