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ON TURAN'S INEQUALITY FOR ULTRA-SPHERICAL POLYNOMIALS
K. VENKATACHALIENGAR and s. K. LAKSHMANA RAO
0. Introduction. Some years ago P. Turan noticed the interesting
inequality
(0.1) An(X) = [Pn(x)]2 - Pn+l(x)Pn-l(x) £ 0, » £ 1, | X \ g 1,
where P„(x) is the Legendre polynomial of order n and the above
inequality has been extended in recent years to the case of some other
orthogonal polynomials as well as the ordinary and modified Bessel
functions [l; 2]. B. S. Madhava Rao and V. R. Thiruvenkatachar
[3] deepened the inequality (0.1) by showing that
d2 2 T d I2(0.2) — A„(x) = - - Pnix) Z 0.
dx2 n(n + 1) Ldx J
The inequality (0.1) has also been generalized to the case of ultra-
spherical polynomials P£M(x) in the form
(0.3) A,.x(«) = [FnAx)]2 - Fn+i.x(x)Fn-i,x(x) ^ or < 0,
according as \x} ^ or >1, where £n,x(x) =PnMix)/Pnx\l). V. R.
Thiruvenkatachar and T. S. Nanjundiah [4] and A. E. Danese [5]
have obtained series of positive functions for A„(x) and its analogue
in the case of the ultraspherical, Laguerre and Hermite polynomials.
In the present paper we deepen the above results on ultraspherical
polynomials by determining the signs of the first and second deriva-
tives of AnX)ix)=[PnX)ix)]2-Pn+1ix)Pnx!.lix) for various values of X
and x. We notice first of all that dAnX)ix)/dx can be represented
as a numerical multiple of the Wronskian Pn+iix)dPnxllix)/dx
—Pn-i(x)idPn+lix)/dx) and express it also via the Christoffel-
Darboux formula as a series of functions of constant sign. The sign
of dA!nX)ix)/dx for various values of X and x follows easily from this.
We determine the sign of (X — l)xidA„)ix)/dx) for a general value of
n and X and show that id2/dx2)AnX)ix) ^0 according as X>1 or
1/2 ^X<1, thus extending the result of B. S. Madhava Rao and
V. R. Thiruvenkatachar to the case of ultraspherical polynomials.
For integer values of n we obtain a series of positive functions for
id2/dx2)Anx\x). Next we show that (1-x2)AlM(x) decreases steadily
in (0, 1) when KX^3/2. From the signs of dA^ix)/dx,
Received by the editors August 9, 1956 and, in revised form, April 1, 1957.
1075License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
1076 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December
d2AnX)(x)/dx2 we prove the concavity of A„,x(x) for 0 <X^ 1/2 and also
arrive at inequalities for An,\(x) in the range \x\ :£1, which are not
only simpler but also cover a wider range of X than the corresponding
inequalities given by O. Szasz [2]. Finally we determine an integral
representation for An(x).
1. Expressing dA^\x)/dx as a Wronskian. We append below the
well known relations satisfied by the ultraspherical polynomials [6]
for frequent use in the following.
d2y dy m(1.1) (1 - x2) — - (2X 4- l)x— + 11(11 + 2X)y = 0, y = P„ (x)
dx2 dx
(1.2) nPn\x) - 2(m + X - l)xPnX-i(x) + (n + 2X - 2)Pn-i(x) = 0,
w = 2
(1.3) -P»' (*) = /(r(X + r)/r(X))P(„"!;) (*), r ^ ndxr
ex) d (\) o (X)(1.4) nPn (x) = x—Pn (x)-Pn-i(x),
dx dx
(X) d (x) d a)(1.5) (n + 2X)Pn (X) = — Pn+l(x) - X — Pn (x),
dx dx
(xi d <x) d (x>(1.6) 2(n + X)P„ (x) = — Pn+i(x)-P„-i(x),
dx dx
(1.7) (1 - x2) — Pn\x) = - nxPT(x) + (n+2\- l)P»_i(*),
(1.8) (1 - x2) — Pn\x) = (» + 2X)*P,<iX>(.T) - (« + l)P„(+i(*).
Differentiation of
(1.9) A? \x) = [PlX,(x)]2 - P%i(x)P?2i(x)
yields
(1.10) — An\x) = af\x) - Bn\x)dx
where
„M(.) = P(:\x) ± P?(X) - PZ(X) y Pn+\(X)dx dx
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i957l ULTRASPHERICAL POLYNOMIALS 1077
and
»). , rx) . , d (X) (X) d <x>8n (X) = Pn+l(x) — Pn-l(x) - Pn (x) — Pn (x).
dx dx
We have then
Pn+i(x) Pn-i(x)
(1.11) an (x)+8n (x) = d (x) d (X) .— Pn+l(x) — Pn-l(x)dx dx
Also
nan (x) — (n + 2X)/3„ (x)
= 2(n + \)PnX\x) — Pn\x) - nPnli(x) — P™i(x)dx dx
(X) d (x)- (n + 2\)Pn+i(x) — P„_i(*)
dx
= 2(« + X)P^X)(^) - Pn\x) - (n +2\- l)P™i(x) - Pn+\(X)dx dx
. (X) , „ d (x> , s- (n + l)Pn+i(x) — Pn-i(x)
dx
- (2X - 1) \Pn+\(x) — Pn-l(x) ~ Pnll(X) — P„+\(x)\ .{ dx dx )
The first three terms together are found to vanish on using (1.4),
(1.5), (1.6) and we get
nan\x) - (n + 2\)8n\x) = - (2X - 1)[«?'(*) + Bn\x)\,
so that there follows the relation
(1.12) (n + 2X - l)aT(x) = (n+ l)^X>(x).
Solving (1.11) and (1.12) for anX)(x), 8„)(x) and using (1.10), we have
Pn+l(x) Pn~l(x)d (X) 1 — X
(1.13) — An (x) = ——- d (x) d a, .dx n + X — Pn+i(x) — P„_i(at)
ax ax
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1078 K. VENKA'I ACHALIENGAR AND S. K. LAKSHMANA RAO [December
From (1.3) and (1.13) we can also express the first derivative of
(X) r dr (M ~12 dr (>,) dr (xi
Ankx) m —- Pn \X) - — PUl(x) — Pnll(x)l_dxr J dxr dxr
as a constant times the Wronskian of P„_+r+i(x) and P^X-iix).
2. Series of functions of constant sign for dAnx\x)/dx. Using (1.2)
in succession we can write
(2.1) Pn+\(X) = anXPnll(x) + KP^x) + CnP^x),
where
4(« + X)(w + X- 1)an = -'
nVn + 1)(2.2)
(« + X)(»+2X-2)(» + 2\-3)
nin + 1)(» + X - 2)
Hence the determinant
t/ ^ P„+\(x) P^ix)Ux, y) = ex) (x)
Pn+iiy) Pn-iiy)
can be written in the form
anX Pn-lix) + bnPn-lix) + CnPn-iix) Pn-\(x)
anyPnliiy) + bnPn-i(y) + cnPn-ziy) P™i(y)
.2 2 (X) (X)= an(x — y )Pn-i(x)Pn-i(y) — cnon-i(x, y)
and so we get2 2 (X) (X)
(2.3) Bnix, y) + cjn-i(x, y) = o„(x - y )Pn_1(x)Pn_1(y).
On observing that
PB+i(*) Pn-i(x)Kix^ti
v-*x y - x — Pn+iix) — P»_i(x)dx dx
we obtain from (2.3), the relation
(n + X) d (X) n-2 + \ d a) m-A„ (x) + cn- A„_2(x) = - 2aBx[PB_i(x)J2,
1 — X dx 1 — X dx
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19571 ULTRASPHERICAL POLYNOMIALS 1079
which can be rewritten in the form
m^ ''.Wn (»+2X-2)(« + 2X-3) d (X)(2.4) — A„ (x)-——- — An_2(x)
dx n(n + 1) dx
8(n + X - 1) r Cx) l2= - \ ^n (1 - \)x[Pn2i(x)]\
n(n + 1)
Solving this difference equation for dA(n)(x)/dx we obtain
d (X), N 8(X- l)r(n+2X- 1)— A„ (x) = -xdx (n + 1)!
[(-+0/21 (n-2k+ 1)!(» - 2k + X 4- 1) w ,
' ^ -w-,lxii± n- l^»-»+i(*)J Ii_i r(» — 2k + 2X + 1)
which is the desired series expansion. Differentiation of (2.5) gives a
series for (d2/dx2)Anx\x) which has however no particularly elegant
form. In the case of Legendre polynomials (X = l/2), (2.5) becomes
& —2x K»+l>/2]
— An(x) = ——— Y (2n - U + 3)[Pn-2k+i(x)]2.dx n(n + 1) t»i
Differentiating this we see that
d2
— 2 I(n+l)/2]
=- E (2» - Ik + 3)P„_24+i(x)[P„-tt+i(*) + 2xP^2*+,(x)]n(n + 1) t-i
which is the same as identity (0.2). From (2.5) it follows that
(2.6) sgn | — a1X>(x)1 = sgn (X - 1)* (X > 0)
which amounts to the following property of Anx'(x).
The Turdn expression AjX)(x) is an increasing (a decreasing) function
for positive (negative) values of x when X > 1. This is to be reversed when
0<X<1.From this we see that
when X > 1, A„ (x) S AB (0) > 0 for all values of x;
when 1/2 ^ X < 1, a1X>(x) S A?°(l) SO for | x| g 1;
when 0 < X g 1/2, C\x) ^ A^\l) ^ 0 for | *| gj 1.
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1080 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December
3. Sign of (X— l)xdA^ (x) / dx for a general index n. For a general
index n (which will be restricted in a way later on) we take P„\x) as
a solution of the differential equation (1.1) which remains regular at
x= +1 (or at x= — 1) either of which is a regular singular point of
the differential equation. We stipulate that the value of the function
as well as its slope at x= +1 are positive. Defining Aj,X)(x) as in (1.9)
we can again derive the relation (1.13) as before. Multiplying this
last relation by (1— x2)XH/2 and differentiating the result, we get
^-\a - xTm t a:\x)]dx L dx J
1 _ X Pn+l(x) Pn-l(x)
= ^r~X~ 1[(1_,Y^P^m] ±\(l-xT1,2yP^(x)VdxL dx J dx\_ dx J
Simplifying the second row of the determinant by means of the
differential equation we are led to
d Y 2 X+l/2 d (X) "I— (1 - x) — An (x)
(3.1) dxl dx J
= 4(l-X)(l-xY-1/2P„X+>1(x)PBX_)1(x).
Hence the extrema of (1 —x2)x+ll2dAnX)(x)/dx occur only at the zeros
of Pn+i(x), Pnxii(x). The solution P„X)(x) as defined above cannot have
zeros on the real segment (1, oo). It increases from a positive value
at x= 1. If it attains a maximum value at a point Xi (> 1), we have
the conditions (dy/dx)xi = 0, (d2y/dx2)zl<Q and from the differential
equation we notice a contradiction if ra(w + 2X) is positive. Hence
P„X)(x) is increasing in x> 1 and consequently does not vanish for any
x on (1, oc). Let us denote by
(a): C*l, 6*2, «3, • • •
03): 0,, 0,, ft, ■ • •
the zeros in ( —1, +1) of P„+i(x) and Pfii(x) respectively. From
(1.13) we have
sgn — An (x) \ = sgn —— PB_i(x) — PB+i(x)LdX JI=a LW + X dX Jx=a
From (1.7) we may write
(I -a2) T^- Pn+\(x) 1 = (n + 2X) PnX\a)LdX J x=a
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i957l ULTRASPHERICAL POLYNOMIALS 1081
and hence
T d (X,, 1 T(X - 1)(» + 2\)PHX!i(a)Pn)(a)lsgn — A„ (x) = sgn -- .
Lax Jx=a L (w + X)(l - a2) J
Using the relation 2(n+\)aPnX)(a) = (n + 2\-l)Plnx±1(a) which follows
from (1.2), we get
d (X) r nsgn — A„ (x) = sgn [(X - l)(n 4- 2\)(n + 2X - l)/a].
In a similar way we observe that
sgn [ — Af'(x)l = sgn [(X - l)nft/(» + !)]•Lax JI=,(j
Combining the above two facts we see that whenever n S1 and
X>0
[d >\) "Ix—A„ (x) = sgn (X - 1).
dx J z=a,S
Thus at all possible extrema of (1-x2)x+1/2(a*A^)(x)/a,x),
Q^ — l)x(dAnX)(x)/dx) is positive and hence
(3.2) sgn (X - l)x — An \x) = + 1, for n S 1, X > 0.
For positive integer values of n this is the same as (2.6).
4. Sign of (d2/ox2)A;,X)(x). Differentiating (1.13) which also holds
for general n we get
D(X) I \ D(X) / \Pn+l(x) Pn-l(x)
d a), s 1 — X-A„ (x) =- d2 (x) d2 (x)dx2 « + X —- Pn+l(x) --P„_l(x)
dx2 dx2
Replacing the second row elements by means of the equations
d1 ex) d2 (xi d ai— Pn+i(x) = X — Pn (x) + (n + 2X + 1) — Pn (*),ax2 dx2 dx
d2 (x> d2 tx) d c\\— p„_;(x) = x — p„ '(*) - («-1) - pV(x)dx2 dx2 dx
(these follow on differentiating (1.5) and (1.4)) and replacing the
first row elements by means of (1.7) and (1.8) we getLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
1082 K. VENKATACHAL1ENGAR AND S. K. LAKSHMANA RAO (December
d2 (X), 1 - X- A„ (x) = -dx2 n + X
(n+2X)x.PnX)(x)-(l-x2) — pT(x) «xPnX)(x) + (l-x2) — P*\x)dx dx
n + 1 n + 2X - 1
d2 rx) d (xj d2 (x) d (X)x — P\ \x) + (ii+2\+l) — Pn \x) x--Pn\x)-(n-l) — P:\x)
dx2 dx dx2 dx
a form in which the right hand side involves only algebraic combina-
tions of P„x>(x) and its first and second derivatives. After simplifying
this determinant we arrive at the equation
(»+l)(«+2X-l) d2 (x, . Vd (x, "j2-An (x) = 2 — P„ (x)
2(X - 1) dx2 \_dx J
(4.1) + (2X - l)x |*[(£ PnX)(x)J - Pn(X,(x) ~ Pf (*)]
-p?\X)JLp?\X)\.dx )
Ii
(4.2) D?(x) = A^x) = \- PlX)(x)]2 - - pV+i(x) - P» (,),Lax J ax dx
we can easily see that DnX)(x) =4X2Anx_+i')(x) and
yD?\x)dx
= 2x{x[(£p1X)(x))2 - PIX)(X) ^PlX)(x)] - A)-ip.WW} •
We may therefore write (4.1) in the form
(»+l)(»4-2X-l) d2 w-An (X)
2(X - 1) dx2(4.3)
T d rx) "I2 d (x+D= 2 - p; \x) + 2x(2x - i)x - a;j; (*).
Lax J dx
Either of (4.1) and (4.3) is an extension of (0.2) to the case of ultra-
spherical polynomials. When n is a positive integer we may replace
the last term on the right side of (4.3) by means of the series (2.5)
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i957l ULTRASPHERICAL POLYNOMIALS 1083
and so arrive at the following series for d2ABx>(x)/dx2
(»+l)(»+2X- 1) d2 (x,/ N-■-A„ (x)
2(X - 1) dx2
r d (x) "I r(« -I- 2X)= 2 — PB \x) \2 + (2X - 1) —- 16X2x2
\_dx J nl
"»™1 in - 2k) l{n - 2k + X + 1) u+1)• 2^ -:-:— LPB-2fc(x)J-.
£[ Tin - 2k + 2X + 2)
From (3.2) it follows that \xdAn\\l)ix)/dx>0 for w^2, X>-l/2.
Using this in (4.3) we deduce that
(a « d* a(XV ^ /+1 (X>1)'(4.5) sgn-AB (x) = <
B dx2 l-l (1/2 |X < 1).
Hence ABX)(x) is a convex function of x for X> 1 and a concave func-
tion of x for l/2gX<l.
5. Decreasing nature of (1 — x2)A„X)(x). From (3.1) we know that
A„X)(x) satisfies the differential equation
r d'2 d i cm(5.1) |^(1 - x2) — - (2X + l)x - + 4(1 -X)jAB (*)
= 4(l-X)[P:X)(x)]2.
If (1 —x2)AnX)(x) is denoted by y, we get
— = (1 - x ) —- An (x) - 2xA„ (x),dx dx
(5.2)
-= (1 - x ) -A„ (x) — \x— AB ix) — 2AB (x).. dx2 dx2 dx
We find from (5.1) and (5.2) that y satisfies the differential equation
(5.3) [(1-^£i-(2X-3)^x-(4X-6)]^
= (4X - 6)x2ABM(x) + 4(1 - X)(l - x)[Pn\x)]\
Now y(0)=A;X)(0) is positive when X>0 and y(l)=0. When X>1,
dy/dx is negative for all x > 1 and hence y decreases for x > 1 when
X> 1. We shall now see that y steadily decreases even in (0, 1) when
KXg3/2. On using (2.6) we have idy/dx)x=o = 0. From (5.3) we
have (d2y/ix2),=o = (4X-6)AiX)(0)-r-4(l-X)[PiX)(0)]2 and so when
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1084 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December
l<X|3/2, id2y/dx2)x=o is negative. Hence when KX|3/2, v reaches
a maximum at x = 0 with the maximum value =ABX)(0) and decreases
in the neighbourhood of x = 0. Also (dy/dx)x=i= — 2A„X)(1) is negative
when X>l/2 and so y(x) is decreasing in the neighbourhood of
x= 1 also when X> 1. If y(x) is not steadily decreasing in (0, 1) it must
reach a minimum value at some point x = a<l, then increase to a
maximum at some point x=j8 between a and 1 and then decrease
again. We show that this possibility cannot occur when 1 <X|3/2.
From the conditions for minimum, we have at x = a, (dy/dx) =0 and
(d2y/dx2)>0 and from the differential equation (5.3) we get
2 / d2y\ i m(1 - a ) ( ~ ) = (4X - 6)yia) + (4X - 6)a AB \a)
\ dx2/x=a
+ 4(l-X)(l-a2)[PBX>(a)]\
The left hand side is positive while the right hand side is negative.
We conclude that y cannot have a minimum anywhere in (0, 1).
Therefore v(x) must steadily decrease in (0, 1) when 1 <X|3/2. It is to
be expected that for large values of X, y(x) decreases first to a mini-
mum and then attains a maximum in (0, 1) before decreasing again
at and near x = 1.
Eliminating (1— x2)d2AnX)(x)/tix2 in d2y/dx2 by means of (5.1) we
obtain
— = (2X - 3)x— ABX)(x) + (4X - 6)ABX)(x) + 4(1 - X)[P„X)(x)]\dx2 dx
Using (3.2) we notice that sgn d2y/dx2= —1 when 1 <X|3/2.
6. Inequalities for AB,x(x). Let
A„.x(x) = [p1X)(x)/PbX)(1)]2 - [P(n+\ix)/Pn+\il)] .[P™iix)/P™(l)].
From the following well-known relations (see (4), (5))
2 (X1 ''
,A n A / \ (1 ~ x)Dn (X) 4X \ <M-1>, .(6.1) AB,x(x) =-= -— (1 - x )An_i ix)
where ^B,x = n(w + 2X) [PBX)(1)]2, and the conclusions of the previous
section, we have
(a) A„,x(x) is steadily decreasing in (0, 1) for 0 < X | 1/2;
d2(b) sgn-A„,x(x) = - 1 when 0 < X | 1/2,
dx2
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1957l ULTRASPHERICAL POLYNOMIALS 1085
showing the concavity property of A„,x(x). From (a) we can im-
mediately deduce the well known inequality viz., the Turan expres-
sion A„,x(x)>0 in |x| <1 when 0<Xgl/2. From (6.1) and the con-
cluding results of §(2) we can prove Turin's inequality in full for
X>0.We have seen in (3.2) that
(3.2) sgn|(X- l)x—A„X)(x) = + 1 for n S 1, X > 0L dx J
and in (4.5) that
d2 (x) (+1 XI>I1(4.5) sgn-An (x) = < n S 2.
dx2 l-l 1/2 S X < 1
From the identity D„X) (x) = 4X2A„xJt"11) (x) it therefore follows that
d rx)(6.2) sgn Xx — Dn (x) = +1 (n S 2, X > - 1/2),
These show the monotonic increasing-decreasing behaviour and the
convexity-concavity property of D„X)(x) ior various values of X, and
enable us to write the following chain of inequalities: When
X > 0, Z?„X)(0) ̂ £>lX>(x) ^ A!X)(0)(1 - x/a) + DnX\a) x/a
^ Dnk\a),
when
-1/2 < X < 0, D„X,(0) ̂ DnX\x) S DnX)(0)(l - x/a) + D™(o) x/a
S DnX\a) 0 ^ x ^ a.
Taking a = 1 and multiplying the above inequalities by (1 —x2) we get
the following inequalities for A„,x(x) in the range |x| ;S1, for w^2.
A„,x(0)(l - x2) =S A„,x(x) ̂ rAn,x(0)(l - x) + ——— (1 - x2)L 2X + 1_
= (1 " x2)/(2\ +1) (X > 0),(6.4)
^fri = l^TTl + A».x(0)(l - *)} (1 - x2) g An.x(x)2X + 1 (2X + 1 )
g ABlX(0)(l - x2) (-1/2<X<0).
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1086 K. VENKATACHALIENGAR AND S. K LAKSHMANA RAO [December
These are simpler than the inequalities for A„,x(x) in O. Szasz's paper
[2] and are at the same time an improvement over the corresponding
ones in [4].
7. Integral representation for A„(x). From (3.1) we have
(7.1) ±\il - x2)^ L AlX)(x)l + 4(1 - X)(l - *tX'\Vix)dx L ox J
= 4(1 - X)(l - x2)X-1/2[P„lX)(x)]2.
If we denote [(l-x2)*-1'2AnK)(x)]/x by/„X)(x), then
(1 - x2)X+m~ A«\x) = £ [x{l _ J)f?\x)] + (2X + l)x2f*\x).dx dx
Hence from (7.1) we get
— |"»(1 - *2)Ax)l + (2X + l)x2 ~fn\x) + 6x/„X>(x)dx2 L J dx
, , 2vX-I/2r (X) , . -,2
= 4(1-X)(1-X) [Pn \x)] ,
which may also be written in the form
d T 2 2 -x+3/2 d (X) "1 rr,(X)/ \i2— x (1 - x ) —/„ (x) = 4(1 - X)x[Pn (x)J .dx L dx
For Legendre functions (X = l/2) this becomes
(7.2) -f-^(l - *2) — [*»(*)/*]) = 2x[Pn(x)]2.dx \ dx /
Hence x2(l —x2)(d[AH(x)/x]/dx) is an increasing (a decreasing) func-
tion for positive (negative) values of x and vanishes at x= +1. From
(7.2) we have in succession
~ (An(x)/X) = 2 Ct[Pnit)]2dt,dx x2(l — x2) J 1
and
(7.3) A„(x) = 2x r U Cl[Pnit)]*dt,J 1 «-(l — u-) J 1
the latter giving a positive integral representation for A„(x). Using the
relation
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i957) ULTRASPHER1CAL POLYNOMIALS 1087
T (1-X2)[^-Pn(x)]2d , ^ Lax J . ,
- (l-x2)[P„(x)]2 +-——- = - 2x[Pn(x)]2,dxL n(n + 1)
we may also write A„(x) in the form
f r d f}(l-l2)\- Pn(t)\
C r n \-dt J(7.4) A„(x) = x [Pn(t)]2 +-;-\r2dt.
J x { n(n + 1)
References
1. G. Szego, On an inequality of P. Turin concerning Legendre polynomials. Bull.
Amer. Math. Soc. vol. 54 (1948) pp. 401-405.2. O. Szasz, Inequalities concerning orthogonal polynomials and Bessel functions,
Proc. Amer. Math. Soc. vol. 1 (1950) pp. 256-267.3. B. S. Madhava Rao and V. R. Thiruvenkatachar, On an inequality concerning
orthogonal polynomials, Proceedings of the Indian Academy of Sciences, Sect. A vol.
29 (1949) pp. 391-393.4. V. R. Thiruvenkatachar and T. S. Nanjundiah, Inequalities concerning Bessel
functions and orthogonal polynomials, ibid. Sect. A vol. 33 (1951) pp. 373-384.
5. A. E. Danese, Explicit evaluations of Turin expressions, Annali di Matematica
Pura ed Applicata, Serie IV vol. 38 (1955) pp. 339-348.6. G. Szegb, Orthogonal polynomials, (1939) pp. 80-84.
Mysore University and
Indian Institute of Science
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