On approximation of ultraspherical polynomials in the oscillatory region
aa r X i v : . [ m a t h . C A ] A ug ON APPROXIMATION OF ULTRASPHERICAL POLYNOMIALSIN THE OSCILLATORY REGION
ILIA KRASIKOV
Abstract.
For k ≥ α ≥ − (2 k + 1) /
4, we provide a uniformapproximation of the ultraspherical polynomials P ( α, α ) k ( x ) in the oscilla-tory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. We show thatthere the function g ( x ) = c p b ( x ) (1 − x ) ( α +1) / P ( α,α ) k ( x ) = cos B ( x ) + r ( x ),where c = c ( k, α ) is defined by the normalization, B ( x ) = R x b ( x ) dx , and thefunctions c, b ( x ) , B ( x ), as well as bounds on the error term r ( x ) are given bysome rather simple elementary functions. Keywords: orthogonal polynomials, ultraspherical polynomials, Gegenbauerpolynomials, uniform approximation Introduction
The ultraspherical polynomials we deal with in this paper will be convenient todefine in terms of Jacobi polynomials as P ( α, α ) k ( x ), where we choose the standardnormalization for the last function. We will use the bold character P ( α, α ) k ( x ) todenote the orthonormal Jacobi polynomials. Since we are going to consider thecase α ≤ − (cid:12)(cid:12)(cid:12)(cid:12) P ( α, α ) k (cid:12)(cid:12)(cid:12)(cid:12) L = Z − (1 − x ) α (cid:16) P ( α, α ) k ( x ) (cid:17) dx = 2 α +1 Γ ( k + α + 1)(2 k + 2 α + 1) Γ ( k + 2 α + 1) k ! , and therefore the orthonormal normalization, still make sense as far as α > − k + 12 .Here we will establish a uniform approximation of the ultraspherical polynomialsin the oscillatory region with an explicit error term for a vast range of the parameter α ; in fact, for all α for which the expression “oscillatory region” makes sense. Afew standard formulas we are using in the sequel may be found e.g. in [13].There are a number of known asymptotics for the Jacobi polynomials underthese or those restrictions on the parameters α and β , starting from the classicalcase | α | , | β | ≤ / α/k and β/k (see e.g. [12], [14] andreferences therein). However if one is interested in uniform bounds the situationbecomes less studied, and we refer to the recent preprint [5] and the referencestherein for a review of known results.To simplify otherwise complicated expressions in the sequel we introduce thefollowing new parameters:(1) u = ( k + α )( k + α + 1) , q = ( α − /u, Mathematics Subject Classification which turn out to be quite natural in this context.We start with the normal form of the differential equation for ultrasphericalpolynomials(2) y ′′ + b y = 0 , y = (1 − x ) ( α +1) / P ( α, α ) k ( x ) , where(3) b = b ( x ) = p (1 − q − x ) u − x , and define the function(4) g ( x ) = p b ( x ) y ( x ) , that, as we will show, almost equioscillates in the interval | x | < √ − q .Our main result is the following Theorem 1 that provides a uniform approxi-mation of g ( x ) for k even in the oscillatory region with a very explicit error term.The corresponding result for k odd may be readily obtained from e.g. the threeterm recurrence. To simplify the statement of the theorem involving multivaluedfunctions, without loss of generality we will restrict ourselves to the case x ≥ Theorem 1.
Let k ≥ be even and let x belong to one of the following intervalsdepending on the value of α :( i) ≤ x ≤ r − u , | α | < r
76 ;( ii) ≤ x ≤ √ − q , α ∈ (cid:2) − k + 14 , − r (cid:3) S (cid:2) r , ∞ (cid:1) . Then the following approximation holds: (5) g ( x ) = g (0) (cid:0) cos B ( x ) + r ( x ) (cid:1) , where (6) g (0) = (cid:18) − (cid:19) k/ (cid:18) k + αk/ (cid:19) (cid:0) k + 2 kα + k + α + 1 (cid:1) / ;(7) B ( x ) = √ u arccos s − q − x − q − √ q arccos s − q − x (1 − q )(1 − x ) ! , q ≥ √ u arccos s − q − x − q + √− q arccosh s − q − x (1 − q )(1 − x ) ! , q < . The error term r ( x ) is bounded as follows: (8) | r ( x ) | < . x p (1 − x ) u , q < − x ) x (1 − q )(1 − q − x ) / √ u , ≤ q < ;(1 + q ) x − q − x ) / √ u , ≤ q < . N APPROXIMATION OF ULTRASPHERICAL POLYNOMIALS... 3
As a corollary we deduce that the ultraspherical polynomials (in a sense also for α ≤ − −√ − q , √ − q ); more precisely Theorem 2.
For k ≥ even, α ≥ − k , | α | ≥ , Z η − η (1 − x ) α (cid:16) P ( α,α ) k ( x ) (cid:17) dx > − − q ) / u / > − · (cid:18) k + α ( k + 2 α ) k (cid:19) / , where η = p − q (cid:18) − · / − q ) / u / (cid:19) . Let us make a few comments about the theorems.
Remark 1.
In fact, the first formula for B ( x ) holds also for q < , provided onechooses the principal branches of the square roots and arccosines. The expressionfor B ( x ) can be slightly simplified by the substitution x = √ − q sin φ , yielding B ( x ) = √ u φ − √ q arccos 1 p q tan φ ! , ≤ φ < π . Remark 2.
For k ≥ and α ≥ − (2 k +1) / the parameter q satisfies the inequalities (9) 4 √ − ≤ − k + k − p ( k − k ( k + 1)( k + 2) ≤ q < . Let us also note that q < corresponds to | α | < , and q = to α ≈ (1 ± √ k + ) . Remark 3.
For orthonormal Jacobi polynomials Stirling’s approximation gives g (0) ≈ ( − k/ r k + 2 α + 1 π . Remark 4.
For α ≥ p / the interval | x | < √ − q is large enough to include allthe zeros of P ( α, α ) k ( x ) . We will show that even at the extreme zeros the error term isstill of order O (1) (see Remark 5 below). It seems not too much is known about thezeros of P ( α,α ) k ( x ) for α < − (see however [2] , [4] ). Nevertheless, for negative α Theorem 1 covers practically the whole oscillatory region inside the interval [ − , besides maybe extreme zeros. Indeed, it’s easy to see that for a continuous function a ( x ) , a nontrivial solution f ( x ) of the differential equation f ′′ + af = 0 may haveat most one zero in each interval where a ( x ) < . For α = − (2 k + 1) / the lengthof interval | x | < √ − q shrinks to approximately √ /k , however for α ≤ − k/ the corresponding ultraspherical polynomial has no zeros in the interval | x | < (see [13, sec. 6.72 ] ). The main term
We will use a version of WKB method presented in [10]. The derivation ofthe approximation (5) is quite straightforward, and some rather technical work isneeded to estimate the error term r ( x ) only.The function g ( x ) satisfies the following differential equation(10) g ′′ − b ′ b g ′ + (1 + ε ) b g = 0 , I. KRASIKOV where(11) ε = ε ( x ) = 3 b ′ − bb ′′ b = − x + 6 qx − x + 4 q − q + 24 u (1 − q − x ) . Solving this equation as inhomogeneous one with the right hand side − εb g, weobtain(12) g ( x ) = M cos( B ( x ) + γ ) + R ( x ) , where(13) B ( x ) = Z x b ( t ) dt, is given explicitly by (7), and(14) R ( x ) = Z x ε ( t ) b ( t ) g ( t ) sin (cid:2) B ( x ) − B ( t ) (cid:3) dt. In the case of ultraspherical polynomials (12) the constants of integration M and γ can be readily found, and we obtain the following claim, which is the first (andeasy) part of Theorem 1. Lemma 3.
For k even, (15) g ( x ) = g (0) (cos B ( x ) + r ( x )) , where r ( x ) = R ( x ) /g (0) , and g (0) = u / (1 − q ) / P ( α, α ) k (0) = (cid:18) − (cid:19) k/ (cid:18) k + αk/ (cid:19) (cid:0) k + 2 kα + k + α + 1 (cid:1) / . Proof.
Plugging x = 0 into (12) yields M = g (0) cos γ . Here the constant γ mustvanish since by (4) g ′ (0) = (1 − q ) / u / ddx P ( α,α ) k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 = 0 , for k even, whereas (15) gives g ′ (0) = − M p (1 − q ) u sin ( B (0) + γ ) + R ′ (0) = − M p (1 − q ) u sin γ. (cid:3) The error term
In what follows we will assume that k is even, and, whenever it is convenient,that x ≥ x belonging to an interval I we will estimate the error term in the followingstraightforward manner:(16) | r ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) g (0) Z x ε ( t ) b ( t ) g ( t ) sin [ B ( t ) − B ( x )] dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ Z x | ε ( t ) b ( t ) | dt, where(17) µ = µ ( k, α ) = sup x ∈I (cid:12)(cid:12)(cid:12)(cid:12) g ( x ) g (0) (cid:12)(cid:12)(cid:12)(cid:12) . N APPROXIMATION OF ULTRASPHERICAL POLYNOMIALS... 5
To estimate sup x ∈I | g ( x ) | we consider the envelope of g ( x ) given by Sonin’s function S ( x ),(18) S ( x ) = g ( x ) + g ′ ( x ) (cid:0) ε ( x ) (cid:1) b ( x ) = g ( x ) + 4 b ( x ) A ( x ) g ′ ( x ) , where A = A ( x ) = 4 b (1 + ε ) = 4 b + 3 b ′ − bb ′′ . Hence(19) g ( x ) ≤ S ( x ) , as far as A ( x ) ≥
0. The location of the maximum of S ( x ) depends on the function b ( x ) only. Indeed, differentiating S ( x ) and using (10) to get rid of g ′′ , we get S ′ = 8 b A A g ′ , A = A ( x ) = 6 b ′ − bb ′ b ′′ + b b ′′′ . Notice also that for k even the point x = 0 is a local maximum of g ( x ), hence(20) g (0) = S (0) . Since we are mainly interested in the sign of A and A , it will be more conve-nient to deal with the following two polynomials instead:(21) A = A ( x ) = (1 − q − x )(1 − x ) u A =4 u (1 − q − x ) + 3 x (1 − qx − x ) − − q )(1 − q ) , and(22) A = A ( x ) = (1 − x ) (1 − q − x ) / x u / A = (1 − q )(1 − q ) − (1 + q ) x . It will also be convenient to introduce the following notion. We call a multivari-able polynomial p ( x ) , x ∈ R n , a P− polynomial if its coefficients are nonnegativeand it has a positive free term. The only property of the P− polynomials we use inthe sequel is that p ( x ) > R n + .In general, the domain { x : A ( x ) > } depends on α and k . As well, to have b ( x ) > α > − k + k +12 k +1 . In what follows weimpose a slightly stronger constraint, namely α ≥ − (2 k + 1) / Lemma 4.
Let k ≥ , then A ( x ) > in the following two cases: ( i ) 0 ≤ x < √ − q , | α | ≥ √
103 ;( ii ) 0 ≤ x ≤ r − u , | α | < √ . Proof.
To establish the first case we substitute x = q (1 − q ) t t into A yielding A ( t, q ) = (1 + t ) − q A = 5 q t + 6 q (1 + q ) t − − q ) t + 4(1 − q ) u + 4 q − . I. KRASIKOV
Then min q A (0 , q ) = 2 − /u >
0. For q > , t >
0, the function A ( t, q ) has theonly minimum at t = t = r − q q , with A = 25 q A ( t , q ) = 25(1 − q ) ( α −
1) + 8 q − q + 4 q − . Thus, A ( x ) > q ≥ /
4. Let now q < /
4, we substitute q = s s ) , s > α = 10 + δ A getting A = 150 s + 280 s + 112 + 25( s + 1)(3 s + 4) δ s ) > . This proves ( i ).It will be convenient to prove ( ii ) for a slightly lager interval | α | ≤ /
4. Todemonstrate the inequality A ( x ) > x dA dx = 1 − qx − x − u (1 − q − x ) < , for 0 ≤ x ≤ p − /u . This can be done with the help of the substitutions x = p − /u √ t , α = 54 · − s s , k = κ + 2 , yielding the P− polynomial − (1 + s ) (1 + t ) u · x dA dx . Hence in the second case u A ( x ) ≥ u A ( p − /u ) = (cid:0) − α ) (3 − α ) (cid:1) u +7 − α ≥ u +7 − α > . This completes the proof. (cid:3)
We need one more technical claim.
Lemma 5. r − u < s (1 − q )(1 − q )1 + q , provided k ≥ and | α | ≤ p / .Proof. For | α | ≤ q ≤ u for 1 < | α | ≤ p / , one finds(1 − q )(1 − q )1 + q − u = 4 q u − qu + q + 1(1 + q ) u ≥ q ) u > . (cid:3) By (22) the maximum of S ( x ) is attained at x = 0 for q ≥ /
4, or, in terms of α , for α / ∈ ( α − , α + ), where(23) α ± = 2 k + 1 ± √ k + 16 k + 496 . N APPROXIMATION OF ULTRASPHERICAL POLYNOMIALS... 7
For q < / S ( x ) is attained either at x = x , where(24) x = s (1 − q )(1 − q )1 + q , or at the endpoint x = p − /u , if | α | < √ /
3, by Lemmas 4 and 5. Tosimplify the statement of the results we use Lemma 5 to restrict the values of x to0 ≤ x ≤ p − /u in a slightly longer interval of the values of α , | α | < p / q < / S ( x ) for q < / Lemma 6.
Let A ( t ) > , A ( t ) > for ≤ t ≤ x , then (25) g ( x ) ≤ S ( x ) ≤ ε (0)1 + ε ( x ) S (0) = 1 + ε (0)1 + ε ( x ) g (0) . Proof.
Starting with the identity S − bA A S ′ = g ≥ , one obtains S ′ /S ≤ A b A = ddx ln 14(1 + ε ( x )) , where 1 + ε ( x ) > A >
0. Integrating from 0 to x we find S ( x ) /S (0) ≤ ε (0)1 + ε ( x ) , and the result follows by (20). (cid:3) Now we are in the position to find the factor µ . Lemma 7. (26) µ ≤ , α ∈ [ − k + 14 , α − ] ∪ [ α + , ∞ ); r α − α − . , α ∈ (cid:0) α − , − r (cid:3) S (cid:2)r , α + (cid:1) ;2(2 − α ) / p α − − α − , | α | < r . Proof.
We will estimate the maximum of the ratio S ( x ) /S (0) ≥ g ( x ) /g (0) ≥ µ . The first case of (26) is just q ≥ /
4, where S ( x ) /S (0) ≤
1. Suppose now that q < / α ∈ (cid:0) α − , − r (cid:3) [ (cid:2)r , α + (cid:1) . Since the assumptions of Lemma 6 are fulfilled for x ∈ [0 , x ), the maximum of S ( x ) is attained at x = x . Hence S ( x ) /S (0) is bounded by1 + ε (0)1 + ε ( x ) = 1 + (1 − q ) (4 − q )2 (25 v (1 − q ) + 8 q − q + 4 q − , I. KRASIKOV where v = qu = α −
1. Here, as easy to check, for a fixed v and 0 < q < /
4, theright hand side is decreasing in q . The choice q = 0 gives0 < ε (0)1 + ε ( x ) < v − α − α − . , and the result follows.Next, let | α | < p / ε (0)1 + ε ( p − /u ) = 2(1 − qu ) (cid:0) − q ) u − q (cid:1) (1 − q ) (1 + u − qu + 8 q u − q u ) < − α ) u − qu + (1 + 8 q u − q u ) u ≤ − α ) α − − α − . (cid:3) Upper bound on R x | ε ( t ) b ( t ) | dt are given by the following lemma. Lemma 8. (27) Z x | ε ( t ) b ( t ) | dt ≤ x p (1 − x ) u , q < − x ) x (1 − q )(1 − q − x ) / √ u , ≤ q < ;(1 + q ) x √ u (1 − q − x ) / , ≤ q ≤ . Proof.
We have − ε ( x ) b ( x ) = ω ( x )4 √ u (1 − x )(1 − q − x ) / , where ω ( x ) = x + 6 qx − x + 4 q − q + 2 . First we consider the case | α | <
1, that is q < . Then ω ( x ) > x ≤ p − /u . Indeed,16 x ω ′ ( x ) = x + 4 qx − α − u + 5 − α u < (4 α − u − u < . Therefore by u ≥ k ≥ | α | ≤ , we have u ω ( x ) ≥ u ω ( p − /u ) = (19 − α + 4 α ) u + 6 α − ≥ − α + 8 α > . Using easy to check inequality ω ( x ) ≤
245 (1 − q − x ) , q ≤ , x ≤ , we obtain Z x | ε ( t ) b ( t ) | dt = 14 √ u Z x ω ( t )(1 − t )(1 − q − t ) / dt ≤ = 65 √ u Z x dt (1 − t ) p − q − t ≤ √ u Z x dt (1 − t ) / = 6 x p u (1 − x ) . N APPROXIMATION OF ULTRASPHERICAL POLYNOMIALS... 9
Let now q ≥ /
2, then ω ( x ) < x < √ − q . Indeed, replacing x by r (1 − q ) s s and setting q = 1 + p − t ) − p ω ( x ) = 5(1 + p ) s + 6(1 + p )(3 + p ) s + 12(1 + 2 p ) s + 8 p > . Hence in this case4 √ u Z x | ε ( t ) b ( t ) | dt = − Z x ω ( t )(1 − t )(1 − q − t ) / dt =(3 − q − x + 2 qx ) x − q )(1 − q − x ) / + arcsin x √ − q − √ q arcsin √ q x p (1 − q )(1 − x ) . Using x ≤ arcsin x ≤ πx , x ≥
0, we convince thatarcsin x √ − q − √ q arcsin √ q x p (1 − q )(1 − x ) ≤ . Therefore for ≤ q < Z x | ε ( t ) b ( t ) | dt ≤ (3 − q − x + 2 qx ) x √ u (1 − q )(1 − q − x ) / ≤ (1 + q ) x √ u (1 − q − x ) / . Finally let 0 ≤ q ≤ /
2, then ω ( x ) can be written as a difference of two positivefunctions, ω ( x ) = ω ( x ) − ω ( x ), ω ( x ) = (1 − q − x )(2 + 3 q − x − qx − x ) ,ω ( x ) = (7 q − q )(1 − q − x ) + 5 q (1 − q ) , and thus | ω ( x ) | ≤ ω ( x ) + ω ( x ) . One finds Z x ω ( t ) + ω ( t )(1 − t )(1 − q − t ) / dt =(9 + 18 q − q − x − qx ) x − q )(1 − q − x ) / − arcsin x √ − q ≤ (9 + 18 q − q − x − qx ) x − q )(1 − q − x ) / ≤ − x ) x (1 − q )(1 − q − x ) / , hence Z x | ε ( t ) b ( t ) | dt ≤ (1 − x ) x (1 − q )(1 − q − x ) / √ u . This completes the proof. (cid:3)
Now Lemmas 7 and 8 infer the following claim which completes the proof ofTheorem 1.
Lemma 9. (28) | r ( x ) | < . x p (1 − x ) u , q < − x ) x (1 − q )(1 − q − x ) / √ u , ≤ q < ;(1 + q ) x √ u (1 − q − x ) / , ≤ q < . Proof.
We have to match the bounds of Lemmas 7 and 8. For q <
0, that is for | α | <
1, we have µ ≤ − α ) / p α − − α − . The maximum of the last expression, even in a lager interval | α | ≤ p / /
15 and attained for α = q −√ . Thus, the numerical coefficient in thiscase is less than · = 2 .
72. For 0 ≤ q < / µ for | α | ≥
1. This yields µ ≤ α = 1. In the lastcase q ≥ / µ ≤ . (cid:3) Remark 5.
The obtained bounds on the error term r ( x ) remain meaningful in asubstantial part of the interval | x | ≤ √ − q . If we set x = p (1 − q )(1 − δ ) , then | r ( x ) | < c k − δ − / , where one can take e.g., c = 2 for ≤ q < / , and c = 1 / for q ≥ / . Sharp upper and lower bounds on the extreme zeros of the Jacobipolynomials were given in [6] and [7] respectively. In the ultraspherical case for thelargest zero x max they are simplified to x max = s k ( k + 2 α + 1)( k + α + 1) − k − α + 1) / (1 + 2 θ )2 k / ( k + 2 α + 1) / (( k + α + 1) − k ) / , where < θ < , α > − , k ≥ . This implies x max < s k ( k + 2 α + 1)( k + α + 1) − k (cid:18) − α + 1) / k / ( k + α + 1) / ( k + 2 α + 1) / (cid:19) , where the first factor does not exceed √ − q , whereas the second one is less than − α / / ( k + α ) / k / . Thus, we get x max < (1 − q ) (cid:18) − α / k + α ) / k / (cid:19) , what together with (8) readily yields | r ( x max ) | = O (1) . Remark 6.
In principle, the bounds on the error term can be strengthen by theiterative substitution of (5) instead of g ( x ) into (14). In particular, for large u , alower bound on r ( x ) can be obtained by estimating the following integral (cid:12)(cid:12)(cid:12)(cid:12) Z x ε ( t ) b ( t ) cos B ( t ) sin [ B ( t ) − B ( x )] dt (cid:12)(cid:12)(cid:12)(cid:12) ≥ N APPROXIMATION OF ULTRASPHERICAL POLYNOMIALS... 11 (cid:12)(cid:12)(cid:12)(cid:12) sin B ( x ) Z x ε ( t ) b ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) Z x ε ( t ) b ( t ) sin [2 B ( t ) − B ( x )] dt (cid:12)(cid:12)(cid:12)(cid:12) . The first integral in a closed form is Z x ε ( t ) b ( t ) dt =(3 − q − x + 2 qx ) x − q )(1 − q − x ) / √ u + 14 √ u arcsin x √ − q − √ qu arcsin √ q x p (1 − q )(1 − x ) . The second one contains a highly oscillating function and probably is negligible incomparison with the first. Thus it seems reasonable to conjecture that the errorbounds of Theorem 1 are of the right order. In fact the situation is slightly moresubtle, because for q < / the main term changes the sign in the interval < x < √ − q . Remark 7.
It would be very interesting to get a uniform bounds on ultrasphericalpolynomials in the transition region. An analogue of (12) (with the cosine replacedby Bessel functions) is readily available (see [11] ). However it is unclear how onecan fix the constants of integration similar to M and γ above. The same problem(but seemingly in a less severe form) arises if we try to extend the results fromthe ultraspherical case to the general Jacobi polynomials. On the other hand, anamusing feature of the method we have used in this paper is that one does not needto know the constants of integration to estimate the relative error r ( x ) . Proof of Theorem 2
We will use the following inequality for the (continuous) central binomial coeffi-cients:(29) (cid:18) xx (cid:19) = Γ(2 x + 1)Γ ( x + 1) > x q π ( x + ) , x ∈ R + . It is a direct consequence of the following: the function w ( x ) = 4 − x (cid:0) xx (cid:1)q π ( x + )is decreasing in x for x >
0, and, as Stirling’s approximation shows, tends to oneas x → ∞ . To check that the function is decreasing, one finds12 (log w ( x )) ′ = ψ (2 x + 1) − ψ ( x + 1) − ln 2 + 14 x + 2 , where the polygamma function ψ ( x ) satisfies the inequalitiesln x − x − x < ψ ( x ) < ln x − x , x > , (see [1] and [3] for the lower and upper bound, respectively). These imply that(log w ( x )) ′ <
0, we omit the details.
Proof of Theorem 2.
By Theorem 1 we have J = g − (0) √ u Z η (1 − x ) α (cid:16) P ( α,α ) k ( x ) (cid:17) dx = g − (0) Z η g ( x ) dx p − q − x = Z η cos B ( x ) dx p − q − x + 2 Z η r ( x ) cos B ( x ) dx p − q − x + Z η r ( x ) dx p − q − x := J ( x ) + 2 J ( x ) + J ( x ) ≥ J ( x ) − |J ( x ) | . Using B ′ ( x ) = b ( x ) and integrating by parts, we obtain J ( η ) = 12 Z η dx p − q − x + 12 Z η cos 2 B ( x ) p − q − x dx =12 arctan η p − q − η + 14 √ u Z η (1 − x )1 − q − x d sin 2 B ( x ) =12 arctan η p − q − η + (1 − η ) sin 2 B ( η )4 √ u (1 − q − η ) − q √ u Z η x sin 2 B ( x )(1 − q − x ) dx, where (cid:12)(cid:12)(cid:12)(cid:12)Z η x sin 2 B ( x )(1 − q − x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z η xdx (1 − q − x ) = η − q )(1 − q − η ) . This along with the inequalityarctan x > π − x , x ≥ , implies J ( η ) > π − p − q − η η − − q − η + 2 q η − q )(1 − q − η ) √ u . A straightforward bound on |J ( x ) | is |J ( x ) | ≤ Z η | r ( x ) | dx p − q − x = qη (1 − q ) (1 − q − η ) √ u − ln 1 − q − q − η (1 − q ) √ u , ≤ q < ;(1 + q ) η − q )(1 − q − η ) √ u , ≤ q < . Thus, in either case we can take |J ( x ) | ≤ η (1 − q )(1 − q − η ) √ u . Since η < √ − q , this yields J > π − p − q − η η − − q + 7 η + 2 η q − q )(1 − q − η ) √ u >π − p − q − η η − − q − η ) √ u . Let δ = 4 · / − q ) / u / , then η = √ − q (1 − δ ), and noticing that δ < / k ≥
10 and α ≥ − k/
2, one finds p (2 − δ ) δ − δ ) + 2(2 − δ ) δ (1 − q ) √ u < r δ + 43 δ (1 − q ) √ u = 32 / (1 − q ) / u / . Let L = 2 α +1 Γ ( k + α + 1)(2 k + 2 α + 1)Γ( k + 2 α + 1) k ! , N APPROXIMATION OF ULTRASPHERICAL POLYNOMIALS... 13 that is L is (cid:12)(cid:12)(cid:12)(cid:12) P ( α, α ) k (cid:12)(cid:12)(cid:12)(cid:12) L , provided α > − I = √ − q Z − √ − q (1 − x ) α (cid:16) P ( α,α ) k ( x ) (cid:17) dx > g (0) √ u L (cid:18) π − / (1 − q ) / u / (cid:19) = π (cid:0) k + αk/ (cid:1) √ k + 2 kα + k + α + 12 k +1 √ u L (cid:18) − · / π (1 − q ) / u / (cid:19) , where by (29), π (cid:0) k + αk/ (cid:1) √ k + 2 kα + k + α + 12 k +1 √ u L = π (2 k + 2 α + 1) √ k + 2 kα + k + α + 14 k + α +1 p ( k + α )( k + α + 1) (cid:18) kk/ (cid:19)(cid:18) k + 2 αk/ α (cid:19) > k + 2 α + 12 s k + 2 kα + k + α + 1( k + 1)( k + α )( k + α + 1)( k + 2 α + 1) > s k + 2 kα + k + α + 1( k + 1)( k + 2 α + 1) = 1 s √ u + 1 − − q ) u > − − q ) √ u . Setting z = (1 − q ) / u / and noticing that z > k ≥ , α ≥ − k/ I > (cid:18) − z (cid:19) (cid:18) − · / πz (cid:19) > − z > − · (cid:18) k + α ( k + 2 α ) k (cid:19) / . This completes the proof. (cid:3)
References [1] G.D. Anderson, S.L. Qiu,
A monotonicity property of the gamma function , Proc. Amer. Math.Soc., 125 (11) (1997), 3355-3362.[2] K. Driver, P. Duren,
Trajectories of zeros of hypergeometric polynomials F ( − n, b, ; 2 b, z ) for b < − /
2, Constr Approx. 17 (2001) 169-179.[3] A. Elbert, A. Laforgia,
On some properties of the gamma function , Proc. Amer. Math. Soc.,128 (9) (2000), 2667-2673.[4] K. Driver, M.E. Muldoon,
Zeros of pseudo-ultraspherical polynomials , Anal. Appl. 12, (2014)563-581.[5] T. Koornwinder, A. Kostenko, G. Teschl,
Jacobi polynomials, Bernstein-type in-equalities and dispersion estimates for the discrete Laguerre operator ∼ gerald/ftp/articles/DispLagGen.pdf[6] I. Krasikov, On zeros of polynomials and allied functions satisfying second order differentialequation , East J. Approx., 9 (2003) 51-65.[7] I.
Krasikov, On extreme zeros of classical orthogonal polynomials , J. Comp. Appl. Math., 193,(2006), 168-182.[8] I. Krasikov,
An upper bound on Jacobi Polynomials , J. Approx. Theory, 149, (2007), 116-130[9] I. Krasikov,
On Erd´elyi-Magnus-Nevai conjecture for Jacobi polynomials , Constr. Approx. 28(2008), 113-125.[10] I. Krasikov,
Approximations for the Bessel and Airy functions with an explicit error term ,LMS J. Comput. Math. 17 (1) (2014) 209-225. [11] I. Krasikov,
On the Bessel functions J ν ( x ) in the transition region, an explicit error term ,LMS J. Comput. Math., 17(1) (2014) 273-281.[12] A. B. J. Kuijlaars, A. Martnez-Finkelshtein, Strong asymptotics for Jacobi polynomials withvarying nonstandard parameters , Journal dAnalyse Mathematique, Volume 94, Issue 1, (2004)195234.[13] G. Szeg¨o,
Orthogonal Polynomials , Amer. Math. Soc. Colloq. Publ., v.23, Providence, RI,1975.[14] O.Szehr, R. Zarouf,
On the asymptotic behavior of jacobi polynomials with varying parame-ters , arXiv:1605.02509[math.CA].
Department of Mathematics, Brunel University London, Uxbridge UB8 3PH UnitedKingdom
E-mail address ::