A note on extraction of orthogonal polynomials from generating function for reciprocal of odd number

(1)

c

° Indian National Science Academy DOI: 10.1007/s13226-019-0303-1

A NOTE ON EXTRACTION OF ORTHOGONAL POLYNOMIALS FROM GENERATING

FUNCTION FOR RECIPROCAL OF ODD NUMBERS

Gradimir V. Milovanovi´c

Serbian Academy of Sciences and Arts, Beograd, Serbia &

University of Niˇs, Faculty of Sciences and Mathematics, P.O. Box 224, 18000 Niˇs, Serbia e-mail: [email protected]

(Received 7 May 2017; accepted 15 February 2018)

Motivated by a recent paper by Shashikala [Indian J. Pure Appl. Math. 48 (2) (2017), 177-185] on the extraction four sequences of orthogonal polynomials from generating function from recipro-cal of odd numbers, in this note we identify the weight functions and the intervals of orthogonality of these sequences of polynomials. Two of these sequences can be expressed in terms of partic-ular Jacobi polynomials transformed to[0,1], and other two are non-classical polynomials also orthogonal on[0,1].

Key words : Orthogonal polynomials; weight function; recurrence relation; continued fractions, Gauss’s hypergeometric function.

1. INTRODUCTION ANDPRELIMINARIES

Letx 7→ w(x)be a weight function on [a, b],a < b, such that all momentsµk =

R_b

axkw(x) dx,

k∈N₀, exist and are finite, andµ0 >0. Then, there exists a unique sequence of monic polynomials {πn(x)}∞n=0orthogonal on[a, b]with respect to this weight function, i.e.,

(πn, πm) =

Z _b

a

πn(x)πm(x)w(x) dx=kπnk2δn,m,

whereδn,mis Kronecker’s delta. These polynomials satisfy the three-term recurrence relation

πn+1(x) = (x−αn)πn(x)−βnπn−1(x), n= 0,1,2. . . , (1.1)

withπ0(x) = 1andπ−1(x) = 0, whereαn = αn(w)andβn = βn(w) are recursion coefficients.

The coefficientβ0 may be arbitrary, but is conveniently defined byβ0 =µ0 = R_b

(2)

The same recursion coefficientsαkandβkappear in the so-called Jacobi continued fraction as-sociated with the weight functionw,

F(x) =

Z _b

a

w(t)

x−tdt∼ β0 x−α0−

β1

x−α1− · · · ,

which is known as the Stieltjes transform of the weight functionx7→w(x)(for details see [1, p. 15], [5, p. 114], [6]). For then-th convergent of this continued fraction, we have

β0 x−α₀−

β1 x−α₁− · · ·

βn−1 x−α_n₋₁ =

σn(x)

π_n(x), (1.2)

whereσn(x)are the so–called associated polynomials, defined by

σn(x) =

Z _b

a

πn(x)−πn(t)

x−t w(t) dt, k≥0.

These polynomials satisfy the same three-term recurence relation (1.1), i.e.,

σn+1(x) = (x−αn)σn(x)−βnσn−1(x), n≥0, (1.3)

only with starting valuesσ0(x) = 0,σ−1(x) =−1(cf. [5, pp. 111-114]).

Recently Shashikala [8] has considered the series with coefficient as reciprocal of odd number,

T(x) = 1 +1 3x+

1 5x

2₊_{· · ·}₊ 1

2n+ 1x

n₊_{· · ·}_, _(1.4)

which has the following representation

T(x) =2F1 ³₁

2,1; 3 2;x

´ ,

where

2F1(a, b;c;x) =

∞

X

k=0

(a)_k(b)_k (c)k

xk

k!

is the Gauss hypergeometric function,(a)0= 1,(a)k=a(a+ 1)· · ·(a+k−1) = Γ(a+k)/Γ(a),

is Pochhammer’s symbol, andΓ(z)is Euler’s gamma function.

Using the regularC-fraction of (1.4) (cf. [4]),

T(x) = 1 1+

−1₃x

1+ −₁₅4x

1+

−₃₅9 x

1+ · · ·

−₄_nn22₋₁x

1+ · · · (1.5)

and taking even and odd order convergents of (1.5), Shashikala [8] has obtained four sequences of monic orthogonal polynomials{Q(nν)(x)}n∞=0,ν = 1,2,3,4, which satisfy the three-term recurrence relation (1.1), withQ(₀ν)(x) = 1and

Q(1)₁ (x) =x−1 3, Q

(2)

1 (x) =x−

3 5, Q

(3)

1 (x) =x−

4 15, Q

(4)

1 (x) =x−

(3)

In [8] these polynomials have been denoted byqn(x),sn(x),rn(x),pn(x), respectively, and their

recurrence cofficients are:

α(1)_n = 32n3+ 24n2−1

(4n−1)(4n+ 1)(4n+ 3), β

(1)

n =

(2n−1)2₍₂_n₎2

(4n−3)(4n−1)2₍₄_n_{+ 1)}; (1.6)

α(2)_n = 32n3+ 72n2+ 48n+ 9

(4n+ 1)(4n+ 3)(4n+ 5), β

(2)

n =

(2n)2₍₂_n_{+ 1)}2

(4n−1)(4n+ 1)2₍₄_n_{+ 3)}; (1.7)

α(3)_n = 32n3+ 72n2+ 48n+ 9

(4n+ 1)(4n+ 3)(4n+ 5), β

(3)

n =

(2n)2₍₂_n_{+ 1)}2

(4n−1)(4n+ 1)2₍₄_n_{+ 3)}; (1.8)

α(4)_n = 32(n+ 1)3+ 24(n+ 1)2−1 (4n+ 3)(4n+ 5)(4n+ 7) , β

(4)

n =

(2n+ 1)2₍₂_n_{+ 2)}2

(4n+ 1)(4n+ 3)2₍₄_n_{+ 5)}. (1.9)

First two polynomials,qn(x)andsn(x), are classical orthogonal polynomials (cf. [5, pp.

121-146]) and they have been extracted from denominators, and other two,rn(x) andpn(x), are

non-classical polynomials and extracted from numerators of (1.5).

Remark 1 : There is a mistake in [8, Eq. (10)];r1(x)should bex−4/15(notx−4/5).

According to the previous facts, rn(x) and pn(x) are the associated polynomials. In the next

section we identify their weight functions and the intervals of orthogonality, as well as ones for the classical polynomialsqn(x)andsn(x).

2. WEIGHT FUNCTIONS ANDINTERVALS OFORTHOGONALITY

Since degσn = n−1, very often for monic associated polynomialsˆσn+1(x) we use the notation πn[1](x), i.e.,

π[1]_n (x) = 1

β0 Z _b

a

πn+1(x)−πn+1(t)

x−t w(t) dt, n≥0 (2.1)

(see [5, p. 112]). Then, because of (1.3), we have

π_n[1]₊₁(x) = (x−αn+1)πn[1](x)−βn+1π_n[1]₋₁(x), π[1]₀ (x) = 1, π₋[1]₁(x) = 0, (2.2)

and, according to Favard’s theorem, these monic associated polynomials are orthogonal with respect to some weight functionx 7→ w1(x) on [c, d]. Grosjean [2, 3] developed a theory for finding an explicit expression for the weightw1(x) and a procedure for obtaining its interval of orthogonality

[c, d]⊂[a, b](see also [9], [7], [5, pp. 112-113]). Namely, ifw(x)is a piecewise weight function on

[a, b], then[c, d] = [a, b]and

w1(x) = µ β0w(x)

P.V. Z _b

a

w(t) dt t−x

¶2

+ (πw(x))2

(4)

whereβ0 =µ0= R_b

aw(x) dx and

R_b

aw1(x) dx=β1, and P.V. means Cauchy principal value.

In the sequel we suppose the even weight function on[−1,1], w(−x) = w(x), for which the monic polynomialsπn(x)are even or odd depending on the parity ofn, i.e.,πn(−x) = (−1)nπn(x).

For such polynomials the coefficientsαn in (1.1) are equal to zero for each n. Also, we need the

following result (see [5, pp. 102-103]) for polynomials defined by

p(1)_n (x) =π2n(

√

x) and p(2)_n (x) = π2n+1( √

x) √

x , n= 0,1,2, . . . . (2.4)

Theorem 1 — The sequences of polynomials©p(nν)(x)

ª

n∈N0,ν = 1,2, are orthogonal[0,1]with

respect to the weight functionsw(1)(x) =w(√x)/√xandw(2)(x) =√xw(√x), respectively, and they satisfy the recurrence relations

p_n(ν₊₁) (x) = (x−a_n(ν))p(_nν)(x)−b(_nν)p_n(ν₋)₁(x), n= 0,1, . . . , (2.5)

withp(₀ν)(x) = 1andp(₋ν₁)(x) = 0, wherea(1)₀ =β1,a(2)0 =β1+β2, and forn∈N

a(1)_n =β₂_n+β₂_n₊₁, b(1)_n =β₂_n₋₁β₂_n

and

a(2)_n =β₂_n₊₁+β₂_n₊₂, b(2)_n =β₂_nβ₂_n₊₁.

Now we start with the classical Gegenbauer (or ultraspherical) polynomialsC_nλ(x)orthogonal on

[−1,1]with respect to the Gegenbauer weight functionw(x) = (1−x2)λ−1/2,λ > −1/2(cf. [5,

p. 133]). The corresponding monic polynomialsπn(x) = Cbnλ(x), satisfy the three-term recurrence

relation (1.1), with the coefficients (see [5, p. 132])

αn= 0 (n∈N0), β0= √

π Γ ¡

λ+ 1 2 ¢

Γ(λ+ 1), βn=

n(2λ+n−1)

4(λ+n−1)(λ+n) (n∈N), (2.6)

except the caseλ= 0, whenβ1 = 1/2.

According to Theorem 1, the polynomialsp(1)n (x) = Cb2λn(

√

x)andp(2)n (x) = Cb2λn+1( √

x)/√x

are orthogonal on[0,1]with respect to the weight functions

x7→ (1−√x)λ−1/2

x and x7→ √

x(1−x)λ−1/2,

respectively, and satisfy the recurrence relation (2.5), with the recurrence coefficients

a(1)₀ = 1

2(λ+ 1), a

(1)

n =β2n+β2n+1= 4n

2_{+ 4}_λn₊_λ₋₁

2(λ+ 2n−1)(λ+ 2n+ 1),

b(1)₀ = √

πΓ¡λ+1 2 ¢

Γ(λ+ 1) , b

(1)

n =β2n−1β2n=

n(2n−1)(λ+n−1)(2λ+ 2n−1) 4(λ+ 2n−2)(λ+ 2n−1)2₍_λ_{+ 2}_n₎,

(5)

and

a(2)₀ = 3

2(λ+ 2), a

(2)

n =β2n+1+β2n+2 = 3λ+ 4n

2_{+ 4(}_λ_{+ 1)}_n

2(λ+ 2n)(λ+ 2n+ 2),

b(2)₀ = √

πΓ¡λ+1₂¢ 2Γ(λ+ 2) , b

(2)

n =β2nβ2n+1 = ₄₍_λn(2_{+ 2}n_n+ 1)(₋₁₎₍λ_λ+_{+ 2}n)(2_n₎λ₂₍+ 2_λ_{+ 2}n−_n1)_{+ 1)}.

(2.8)

In fact, they are (monic) Jacobi polynomials transformed to the interval[0,1], with parameters

(λ−1/2,∓1/2), i.e.,

p(1)_n (x) = n! (n+λ)nP

(λ−1/2,−1/2)

n (2x−1) and p(2)n (x) =

n! (n+λ+ 1)nP

(λ−1/2,1/2)

n (2x−1),

wherePn(α,β)(x)are the classical Jacobi polynomials orthogonal with respect to the weight function

x7→(1−x)α_{(1 +}_x₎β _on_[₋₁_,_1]_{(cf. [5, pp. 131-140]).}

Evidently, in the caseλ= 1/2, the coefficients (2.7) and (2.8) reduce to ones for the polynomials {Q(nν)(x)}∞n=0,ν= 1,2, obtained in [8]. Namely,

a(1)_n ¯¯_λ₌₁_/₂=α(1)_n = 8n

2_{+ 4}_n₋₁

(4n−1)(4n+ 3), b

(1)

n

¯ ¯

λ=1/2=βn(1) =

(2n−1)2(2n)2 (4n−3)(4n−1)2₍₄_n_{+ 1)},

a(2)_n ¯¯_λ₌₁_/₂=α(2)_n = 8n

2_{+ 12}_n_{+ 3}

(4n+ 1)(4n+ 5), b

(2)

n

¯ ¯

λ=1/2=βn(2) =

(2n)2(2n+ 1)2 (4n−1)(4n+ 1)2₍₄_n_{+ 3)}.

Notice that the expressions forα(nν),ν = 1,2,3,4, in (1.6)-(1.9), can be shortened by a common

factor.

Thus, we have the following statement:

Proposition 1 — The polynomialsQ(1)n (x)andQ(2)n (x) (i.e.,qn(x)andsn(x))are orthogonal on [0,1]with respect to the weight functionsx 7→ 1/√xandx 7→ √x, respectively, and they can be

expressed in terms of Jacobi polynomials as

Q(1)_n (x) = ¡ n! n+1

2 ¢

n

P_n(0,−1/2)(2x−1) and Q(2)_n (x) = ¡ n! n+3

2 ¢

n

P_n(0,1/2)(2x−1).

We return now to the weight functionx7→w1(x)of the associated polynomial for the Gegenbauer polynomialsC_nλ(x).

Takingt= (z+x)/(1 +xz), we have

P.V. Z ₁

−1

(1−x2₎λ−1/2_d_t

t−x = (1−x

2₎λ−1/2 _P.V. Z ₁

−1

(1−z2₎λ−1/2 z(1 +xz)2λ dz.

Since

(1 +xz)−2λ =

∞

X

k=0 µ

−2λ k

¶

(xz)k=

∞

X

k=0

(−1)k(2λ)k

(6)

and

µk=

Z ₁

−1

zk(1−z2)λ−1/2dz=

   

  

0, kis odd,

Γ¡λ+1 2 ¢

Γ¡k+1 2

¢

Γ¡λ+ 1 +k

2

¢ , kis even,

as well asµ−1 =P.V. Z ₁

−1

z−1(1−z2)λ−1/2dz= 0, we get

P.V. Z ₁

−1

(1−x2)λ−1/2dt

t−x = −(1−x

2₎λ−1/2_Γ µ

λ+1 2

¶_X∞

ν=0

(2λ)2ν+1Γ ¡

ν+1₂¢ (2ν+ 1)!Γ(λ+ν+ 1)x

2ν+1

= −2λβ0(1−x2)λ−1/2x2F1 µ

λ+1 2,

1 2;

3 2;x

2 ¶

,

whereβ0is given in (2.6), so that the weight function (2.3) becomes

w1(x) = β0(1−x 2₎1/2−λ

4λ2_β2 0 ¡

x₂F₁¡λ+ 1

2,12;32;x2 ¢¢₂

+π2, x∈[−1,1].

In the Legendre case(λ= 1/2), it reduces to

w₁(x)¯¯_λ₌₁_/₂= 2

4¡tanh−1x¢2+π2, x∈[−1,1], (2.9)

and the corresponding orthogonal polynomialsπn[1](x)satisfy the three-term recurrence relation (2.2),

with coefficients given by (2.6) forλ= 1/2, i.e.,

π[1]_n₊₁(x) =xπ_n[1](x)− (n+ 1)2 (2n+ 1)(2n+ 3)π

[1]

n−1(x), n= 0,1, . . . . (2.10)

The weight function (2.9) is even andπ[1]n (−x) = (−1)nπ[1]n (x). Applying Theorem 1 we get two

sequences of polynomialspb(1)n (x) =π[1]₂_n(

√

x)andpb(2)n (x) =π₂[1]_n₊₁(

√

x)/√x, which are orthogonal

on[0,1]with respect to the weight functions

x7→wb1(x) = √1_x 2

4¡tanh−1√x¢2+π2 and x7→wb2(x) =

2√x

4¡tanh−1√x¢2+π2,

respectively. Starting from (2.10), i.e.,

b

βn=βn+1 ¯ ¯

λ=1/2=

(n+ 1)2

(2n+ 1)(2n+ 3), n= 0,1, . . . ,

and using Theorem 1, we obtain their recurrence coefficients,ab(nν)andbbn(ν),ν= 1,2, in the following

form

b

a(1)₀ =βb1= ₁₅4 , ban(1)=βb2n+βb2n+1= 8n

2_{+ 12}_n_{+ 3}

(4n+ 1)(4n+ 5),

b_b(1)

n =βb2n−1βb2n=

(2n)2₍₂_n_{+ 1)}2

(7)

and

b

a₀(2) =βb1+βb2 = 11₂₁, ba(2)n =βb2n+1+βb2n+2 = 8n

2_{+ 20}_n_{+ 11}

(4n+ 3)(4n+ 7),

b_b(2)

n =βb2nβb2n+1= (2n+ 1)

2₍₂_n_{+ 2)}2

(4n+ 1)(4n+ 3)2₍₄_n_{+ 5)}.

As before, we can see that these coefficients coincide with those of polynomials{Q(nν)(x)}∞n=0, ν= 3,4, given in (1.8) and (1.9) and obtained in [8]. Namely,

b

a(1)_n =α(3)_n , bb(1)_n =β_n(3) and ba(2)_n =α(4)_n , bb(2)_n =β_n(4),

so that we have the following statement:

Proposition 2 — The polynomialsQ(3)n (x)andQ(4)n (x) (i.e.,rn(x)andpn(x))are orthogonal on [0,1]with respect to the weight functionsx7→wb1(x)andx7→wb2(x), respectively.

Some other interesting cases could be forλ= 0andλ= 1, but they are much simpler than the previous one forλ= 1/2.

In the caseλ= 0we start from the recurrence relation for the monic Chebyshev polynomials of the first kindπn(x) = 21−nTn(x),n≥1,

πn+1(x) =xπn(x)−βnπn−1(x), π0(x) = 1, π−1(x) = 0, (2.11)

whereβ0 =π,β1 = 1/2, andβn = 1/4,n≥2. These polynomials are orthogonal on[−1,1]with

respect to the weight functionx7→(1−x2)−1/2.

An application of Theorem 1 gives two well-known sequences of polynomials orthogonal on[0,1] with respect to the weight functionsx7→1/px(1−x)andx7→px/(1−x),21−2nT2n(

√ x)/√x

and2−2n√x T2n+1( √

x), which satisfy the recurrence relation (2.5), with the coefficients

a(1)_n = 1

2 (n≥0), b

(1)

0 =π, b (1) 1 =

1 8, b

(1)

n = 1

16 (n≥2)

and

a(2)₀ = 3 4 a

(2)

n = 1

2 (n≥1), b

(1) 0 =

π

2, b

(1)

n = 1

16 (n≥1),

respectively.

In the case λ = 1, we start again from the same recurrence relation (2.11), of course, in this case for the monic Chebyshev polynomials of the second kind πn(x) = 2−nUn(x), orthogonal

on [−1,1]with respect to the weight function x 7→ (1−x2)1/2. The recurrence coefficients are β0 = π/2 andβn = 1/4, n ≥ 1. The corresponding sequences of polynomials, 2−2nU2n(

(8)

and2−2n−1U2n+1( √

x)/(√x), obtained by the application of Theorem 1, are orthogonal on[0,1]

with respect to the weight functionsx 7→ p(1−x)/xandx 7→ px(1−x), and they satisfy the recurrence relation (2.5), with the coefficients

a(1)₀ = 1 4, a

(1)

n = 1

2 (n≥1), b

(1) 0 =

π

2, b

(1)

n = 1

16 (n≥1)

and

a(2)_n = 1

2 (n≥0), b

(1) 0 =

π

8, b

(1)

n = 1

16 (n≥1),

respectively.

It is easy to see that the associated polynomials on[−1,1]in the both cases (λ= 0andλ = 1) are the (monic) Chebyshev polynomials of the second kind.

ACKNOWLEDGEMENT

The author was supported in part by the Serbian Academy of Sciences and Arts (No. Φ-96) and by the Serbian Ministry of Education, Science and Technological Development (#OI174014).

REFERENCES

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