revised March 13, 2008 MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS

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1 revised Mrch 3, 008 MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS LEE LORCH AND MARTIN E. MULDOON To the memory of Luigi Gtteschi Abstrct. In the course of their work on Slem numbers nd uniform distribution modulo, A. Akiym nd Y. Tnigw proved some inequlities concerning the vlues of the Bessel function J 0 t multiples of π, i.e., t the zeros of J /. This rises the question of inequlities nd monotonicity properties for the sequences of vlues of one cylinder function t the zeros of nother such function. Here we derive such results by differentil equtions methods.. Introduction As fr bck s 950 [8], Luigi Gtteschi ws interested in the pproximtion of Bessel functions nd their zeros. Lter, he contributed gretly to the use of Bessel functions s pproximnts for other functions; see, e.g., [9, 0] nd references. He often used differentil equtions methods. Since the trigonometric functions re relted to the Bessel functions by J / (x = πx x, J /(x = cos x, πx it is nturl to investigte the question of whether vrious sequences which re constnt in the trigonometric cse might be monotonic in the cse of generl order. Such questions re studied here. They re more elementry thn those delt with by Gtteschi. The element of commonlity is the use of differentil equtions methods. This study is motivted by the result (. J 0 (kπ /(π k, k =,,.... of A. Akiym nd Y. Tnigw [, Lemm ], used in the course of their work on Slem numbers nd uniform distribution modulo. We re deling here with the vlues of the Bessel function J 0 t multiples of π, i.e., t the zeros of J /. This rises the question of inequlities nd monotonicity properties for the sequences of vlues of one cylinder function t the zeros of nother such function. For exmple, we show (Corollry 5.5 tht for 0 ν <, ( k kj ν (kπ increses to its limiting vlue ( /π (( νπ/4 s k (=,,... increses. In prticulr, with ν = 0 this gives (. 0 < ( k J 0 (kπ < π, k =,,..., k generlizing ( Mthemtics Subject Clssifiction. Primry 33C0; Secondry 34C0. Key words nd phrses. Bessel functions, cylinder functions, inequlities, monotonicity properties. This work ws supported by grnts from the Nturl Sciences nd Engineering Reserch Council, Cnd.

2 LEE LORCH AND MARTIN E. MULDOON. Trnsformtions of differentil equtions We consider the differentil eqution (. y + f(xy = 0, < x <, where f is continuous on (,, nd the eqution is nonoscilltory [, p. 35] t. Under these conditions, the eqution hs principl solution t [, p. 355], i.e., solution y (x such tht for every solution y(x linerly independent of y, we hve (. lim x + y (x/y(x = 0. Let y be solution of (. such tht the Wronskin (.3 W (y, y = y (xy (x y (xy (x, nd let (.4 p(x = y (x + y (x, s x. We suppose throughout tht, for fixed c, < c <, (.5 c du p(u = lim ɛ 0 + c +ɛ du p(u <. It is well known tht eqution (. cn be trnsformed to the trigonometric eqution u (t + u(t = 0. The chnges of vribles required re described, for exmple in [3, Lemm.3], where the interest, like here, is in the ppliction to zeros of Bessel functions. Letting y(x = [p(x] / u(t, x (t = p(x (. becomes (.6 u (t + u(t = 0, where the prime now denotes differentition with respect to t. Hence the generl solution of (. my be written s (.7 y(x = A[p(x] / dt p(t + α, where A nd α (0 α < π re rbitrry. The solutions y, y re unique up to their replcement by cos α y (x α y (x, α y (x + cos α y (x in the sense tht this leves W (y, y nd p(x unchnged. So, from here on, we will tke ( (.8 y (x = [p(x] / dt x, y (x = [p(x] / dt cos. p(t p(t We use y(x, α nd y (x, α = y(x, α + π/ for the solutions (.9 y(x, α = cos α y (x α y (x = [p(x] / dt p(t + α, (.0 y (x, α = α y (x cos α y (x = [p(x] / dt p(t + α + π/. The zeros of y(x, α on (, re the numbers x k for which (. xk dt = kπ α, k =,,.... p(t

3 MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS 3 For consistency in nottion, we tke x 0 =, when α = 0. At zero x k of y(x, α we hve (. y(x k, β = ( k p(x k (β α leding to Remrk.. If p(x is monotonic s function of x, so is the sequence y(x k, β. Now we consider second eqution (.3 Y + F (xy = 0, < x <, with similr nottion. Y (x nd Y (x denote linerly independent solutions of (.3 with Wronskin, such tht (.4 P (x = Y (x + Y (x, s x +. The generl solution of (.3 my be written s (.5 Y (x = A[P (x] / dt P (t + α, where A nd α (0 α < π re rbitrry. Without loss of generlity, we my tke ( (.6 Y (x = [P (x] / dt x, Y (x = [P (x] / dt cos. P (t P (t We use Y (x, β for the solution (.7 Y (x, β = cos β Y (x β Y (x = [P (x] / dt P (t + β. The zeros of Y (x, β (on (, re the numbers X k for which (.8 Xk Agin, we tke X 0 =, in cse β = 0. dt = kπ β, k =,,.... P (t 3. Generl results Our min result dels with the vlues of y(x, α t the zeros of Y (x, β. Theorem 3.. Consider the solutions y(x = y(x, α, y (x = y(x, α π/ of (. given by (.9 nd (.0 nd solution Y (x of (.3 with successive simple zeros t x k, x k nd X k (k = 0,,..., respectively on (,. Suppose tht (3. p(x < P (x, < x <, nd tht (3. x k X k x k+, k =,,.... Let y(x > 0. Then ( k+ y(x k / p(x k increses to finite limit s k (=,,... increses. In cse (3. is replced by (3.3 x k+ X k x k+, k =,,..., the conclusion is tht ( k+ y(x k / p(x k decreses to its limit s k (=,,... increses. The monotonicities re reversed if the inequlity sign in (3. is reversed.

4 4 LEE LORCH AND MARTIN E. MULDOON Proof. Becuse of y(x > 0 nd (3., we my use y(x k = ( k+ y(x k. We hve to show tht A k < A k+, k =,,..., where (3. gives A k = Xk du + α, k =,,.... p(u (3.4 kπ A k (k + π, so the A k re in x-intervls where x is increg. Also (3.5 A k+ A k = Xk+ X k du p(u > Xk+ X k du P (u = π, the inequlity following from (3.. From (3.4 nd (3.5, we get (3.6 (k + π A k + π < A k+ (k + 3 π Thus we hve A k+ > (A k + π = A k nd the ssertion bout the increse of ( k+ y(x k / p(x k is proved. In cse (3. is replced by (3.3, the rgument is similr except tht (3.4 is replced by (3.7 (k + π A k (k + π, so the A k re in x-intervls where x is decreg nd (3.6 is replced by (3.8 (k + 3 π A k + π < A k+ (k + π, so the result is A k > A k+. The proof of the ssertion in the lst sentence follows on noting tht when the inequlity in (3. is reversed, the sme hppens to the inequlity in ( Preliminry remrks on zeros of Bessel functions Here we consider the specil cse of eqution (. given by [ (4. y ] ν x y = 0, stisfied by y (x = πx/j ν (x nd y (x = πx/y ν (x, with nd W (y, y =, (4. p(x = p ν (x = π x [ J ν (x + Y ν (x ], s x. We use the usul nottion C ν (x, α = cos αj ν (x αy ν (x, for cylinder functions, nd we use c νk (α for the kth positive zero of C ν (x, α. We lso use the usul nottions j νk nd y νk for the kth positive zeros of J ν (x nd Y ν (x.

5 MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS 5 Á. Elbert nd A. Lforgi [5] (see lso [3] hve shown how to define zero j νκ of continuous rnk κ of C ν (x, α by: j νκ = c νk, where κ = k α π, 0 α < π. So, for exmple, when ν >, j ν,k nd j ν,k, k =,,... give the positive zeros of J ν (x nd Y ν (x, respectively. See lso the discussion nd Figure in [4], which confirms the results of Elbert nd Lforgi [6, Theorem.] tht j νκ is convex function of κ for 0 ν < nd concve function of κ for ν >. It follows from the reltion [] j ν,ν+κ = j ν,κ, ν 0, κ > 0, tht j νκ is lso convex function of κ on its domin of definition for < ν < 0. Lemm 4.. We hve ( (4.3 κ + ν ( π < j νκ < κ + ν 4 4 π, κ 8, for < ν <. The lower bound becomes exct for ν = become exct for ν =. The inequlities re reversed for ν >. nd both bounds Proof. The lower bound in the cse < ν < nd upper bound in the cse ν > re well known nd holds even for κ > 0. (See the work of A. Lforgi [], lso reported in [3]. More precise results re given in [7]. Now j νκ is positive increg function of κ. Further it is convex in κ for < ν < nd concve for ν >. Thus it is sufficient to prove the remining bounds for κ =, tht is ( ν (4.4 y ν < π, 8 < ν <, nd (4.5 y ν > ( ν π, ν > 8. Now if we write ( ν g(ν = y ν π, 8 we hve g (ν = dy ν/dν π/4 which, in view of results in [4] decreses to the positive number π/4, s ν. Thus g(ν is increg, g( = π 4, g( = 0 nd g(ν s ν. This gives the inequlities (4.4 nd (4.5. Wtson [5, pp ] hs this result, but is not explicit bout wht would be, in our terms, the lower bound on κ. 5. Monotonic sequences relted to Bessel functions It is known [5, p. 446] tht p ν (x, s given by (4., is n increg or decreg function of x, 0 < x <, ccording to whether 0 ν < or < ν <. Hence, we hve the following: Remrk 5.. {j / νk Y ν(j νk } nd {y / νk J ν(y νk } re lternting sequences of numbers whose bsolute vlues increse to /π for 0 ν < nd decrese to /π for ν >.

6 6 LEE LORCH AND MARTIN E. MULDOON In ddition, we know [5, p.446] tht p ν (x/x is decreg function of x on (0, for ech fixed ν 0. This shows tht J ν (y νk nd Y ν (j νk form decreg sequences. More generlly, if we hve two cylinder functions of the sme order, C ν (x, α nd C ν (x, β, we hve (5. C ν (x, β = cos(α βc ν (x, α + (α βc ν (x, α π/. If we use {x k } for the zeros of C ν (x, α, we see tht (5. C ν (x k, β = (α βc ν (x k, α π/. In prticulr, we hve, with α = π/, (5.3 C ν (y νk, β = cos(βj ν (y νk nd with α = 0, (5.4 C ν (j νk, β = (βy ν (j νk. Hence the monotonicity results for the specil cses α = 0, π led to similr results in the cse of generl α: Remrk 5.. For ν 0 nd 0 α < π, the sequences C ν (j νk, α nd C ν (y νk, α decrese to 0 except in the trivil cses (α = 0 for the first nd α = π/ for the second where they re identiclly 0. Now we consider second eqution [ (5.5 Y µ x ] Y = 0, with solutions Y (x = πx/j µ (x nd Y (x = πx/y µ (x, with (5.6 P (x = p µ (x = π x [ J µ(x + Y µ (x ]. To compre p µ nd p ν we use Nicholson s integrl representtion [5, p. 444, (] (5.7 p ν (x = 4 π x K 0 (x h t cosh(νt dt. 0 This formul nd relted ones hve been found useful in mny investigtions of monotonicity properties of Bessel function nd their zeros; see [3, ] nd references. In view of (5.7, we hve p µ (x > p ν (x, when µ > ν 0, so Theorem 3. gives: Theorem 5.. Define (5.8 f ν (x = x / C ν (x, α/ p ν (x. Let c µk (β be the kth positive zero of C µ (x, β, where µ > ν 0, nd suppose tht for some positive integer k 0, (5.9 c ν,k+m (α < c µ,k (β c ν,k+m+ (α + π/, k = k 0, k 0 +,..., where m is the integer prt of (α β/π + (µ ν/. Then the sequence f ν (c µ,k is lternting nd f ν (c µ,k increses to its limiting vlue ( (5.0 L = µ ν π π + α β, s k increses from k 0 to. If (5.9 is replced by (5. c ν,k+m+ (α + π/ < c µ,k (β c ν,k+m+ (α, k = k 0, k 0 +,...,

7 MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS 7 then the sequence f ν (c µ,k is lternting nd f ν (c µ,k decreses to L s k increses from k 0 to. The monotonicities re reversed if ν > µ 0. The reson for the choice of m is tht the symptotic formul [5, p. 506] ( (5. c νk (α k + ν π α, k, 4 shows tht in order for (5.9 to hold for some m, it is necessry tht α β + µ ν π < m < α β + µ ν, π nd in order (5. to hold for some m, it is necessry tht α β + µ ν < m < α β + µ ν π π. Here we look t some specific cses of Theorem 5., where k 0 cn be chosen equl to, nd hence we get monotonicity from the strt. Theorem 5.. Suppose tht either (5.3 0 ν < µ, or (5.4 Then (5.5 ( k j / µk J ν(j µk pν (j µk ν < µ 3. ( (µ νπ increses to π = L, (5.6 ( k+ y / µk J ν(y µk pν (y µk decreses to ( (µ νπ π cos = L, (5.7 nd (5.8 ( k+ j / µk Y ν(j µk pν (j µk ( k+ y / µk Y ν(y µk pν (y µk s k (=,,... increses to infinity. 0 µ < ν or µ < ν 3. decreses to increses to π cos π ( (µ νπ ( (µ νπ,, The monotonicities re reversed if either Corollry 5.3. Under the hypotheses of Theorem 5., J ν (j µk /Y ν (j µk nd Y ν (y µk /J ν (y µk increse to the limiting vlue tn ((µ νπ/, s k (=,,... increses. Proofs. The representtion (5.7 gives the condition corresponding to (3., so to check the pplicbility of Theorem 3. we need to verify tht (5.9 j νk < j µk y ν,k+, k =,,... the condition corresponding to (3.. The first inequlity here follows from the increg nture of j νk s function of ν, ν >. The second inequlity cn be expresses s j µκ j ν,κ+ which follows from Lemm 4., provided µ ν + 3

8 8 LEE LORCH AND MARTIN E. MULDOON in cse (5.3 holds, or µ ν + 5 in cse (5.4 holds. These re both esy consequences of the hypotheses on µ nd ν. To get the correct signs, we lso need to show tht J ν (j µ < 0. This follows from the consequence j ν < j µ < y ν of (5.9. Hence (5.5 holds. The result (5.6 is proved similrly, the inequlities (5.9 being replced by (5.0 y ν,k+ < y µ,k+ j ν,k+, k =,,.... Similr remrks pply to (5.7 nd (5.8. The limits of the sequences follow from the symptotic behviour of the Bessel functions nd their zeros [5, Chpter 7]. To see tht the the monotonicities re reversed in cse ν > µ 0, we consider tht in tht cse we hve (5. y νk < j µk < j νk nd (5. j ν,k < y µk < y νk. Corollry 5.3 follows from the Theorem on ug formul (4. for p ν (x. 5.. Cylinder functions of order between nd. Corollry 5.4. If 0 ν < µ then ( k j µk J ν (j µk nd ( k+ yµk Y ν (y µk increse to the limiting vlue L, given by (5.5, s k (=,,... increses. Corollry 5.4 follows from the first prt of Theorem (5. on noting tht, for 0 ν <, p ν(x increses [5, p. 446] to s x increses. Corollry 5.5. For 0 ν <, ( k kj ν (kπ nd ( k+ k Y ν((k π increse to the limiting vlue ( /π (( νπ/4 s k (=,,... increses. In prticulr, with ν = 0, this gives (5.3 0 < ( k J 0 (kπ < π, k =,,.... k nd (5.4 0 < ( k+ Y 0 ((k π <, k =,,.... π k Corollry 5.5 follows from Corollry 5.4 on tking µ =. The consequence J 0 (kπ < /(π k of the left-hnd inequlity in (5.3 ws proved (with for m even in [, Lemm ]. 5.. Cylinder Functions of order between nd 3. Corollry 5.6. Suppose tht ν < µ 3. Then ( k+ yµk J ν (y µk nd ( k+ j µk Y ν (j µk decrese to the limiting vlue L, given by (5.6, s k (=,,... increses. This follows from (5.6, (5.7, ce p ν (x is decreg. Corollry 5.7. Suppose tht < ν 3. Then ( k+ kj ν (kπ nd ( k k Y ν((k π increse to the limiting vlue ( /π cos((ν π/4 s k (=,,... increses.

9 MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS 9 Clerly the condition corresponding to (3. holds so to check the pplicbility of Theorem 3. we need to verify tht (5.5 y νk < j µk < j νk, k =,,.... the condition corresponding to (3.. The second inequlity here follows from the increg nture of j νk s function of ν, ν >. To show the left-hnd inequlity in (5.5, it suffices, in view of Lemm 4., to verify tht kπ 3 4 π + νπ kπ 8 π + 4 µπ, or tht µ ν 5, n esy consequence of the hypotheses on µ nd ν. The result concerning Y ν is proved similrly, the inequlities (5.5 being replced by (5.6 j ν,k < y µ,k+ y ν,k+, k =,,.... The right-hnd inequlity is obvious nd the left-hnd one follows s before. Now we expnd little on the sitution where the monotonicities re reversed. We suppose tht either (5.7 0 µ < ν, or tht, (5.8 µ < ν 3. Then the reversed form of 5.5 refers to sequence of negtive numbers decreg to negtive limit nd it cn be expressed s ( (ν µπ (5.9 increses to π ( k+ j / µk J ν(j µk pν (j µk In the prticulr cse where ν = Corollry 5.7 gives: Corollry 5.8. The positive quntities ( k+ k / J (kπ increse to /π s k (=,,... increses. Hence ( < ( k+ J (kπ < π, k =,,..., k in nlogy to ( Vlues of cylinder functions t multiples of π. The bove results del with situtions where µ nd ν differ by t most. Numericl evidence indictes tht we cnnot expect monotonicity, though we might expect ultimte monotonicity, when µ nd ν re further prt. We confine ourselves here to exmining further the situtions rig in Corollries 5.5 nd 5.7, which del with the vlues of cylinder function t the points kπ, where k runs through sequence of integers. For such sequences, we will be interested in estimting the point t which they become monotonic. Theorem 5.9. Let n < ν < n +, for some positive integer n. In cse n = s + is odd, let (5.3 y νk < (s + kπ < j νk, k k 0,

10 0 LEE LORCH AND MARTIN E. MULDOON for some positive integer k 0. Let s k = k / J ν (kπ/ p ν (kπ, t k = k / Y ν (kπ/ p ν (kπ, k k 0 + s, Then {s k } nd {t k } re lternting sequences, s k increses to ( /π cos((ν + π/4 nd t k decreses to ( /π ((ν + π/4 s k( k 0 + s increses. In cse n = s is even nd (5.3 is replced by (5.3 j νk < (s + kπ < y ν,k+, k k 0, the monotonicities of s k nd t k re reversed. Proof. This is strightforwrd ppliction of Theorem 3. to the equtions (4. nd (5.5 with µ =. The inequlity (3. is reversed nd (3. is replced by (3.3 for the J cse. In the Y cse, we use the sme Theorem with y(x, α = x / Y ν (x, y(x, α π/ = x / J ν (x. Corollry 5.0. Let ν = m + be n odd positive integer nd suppose tht ( ν (5.33 j ν,k > + k π, for some positive integer k. Then nd hence lso s k ( k + m increses. ( k m+ k / J ν (kπ pν (kπ ( k m+ k / Y ν (kπ pν (kπ increses to /π, decreses to /π, ( k m+ k / Y ν (kπ decreses to /π, Proof. We hve to verify tht the conditions (5.3 re stisfied. The left-hnd inequlity is the consequence y νk < k + ν/ 3/4 of Lemm 4.. In view of the concvity of j νκ s function of κ, the inequlity j νκ > (m + kπ for κ = k (ssumption (5.33, will show it to be true for ll κ > k. For positive even integer vlues of ν, we get: Corollry 5.. Let ν = m be n even positive integer nd suppose tht ( ν (5.34 y ν,k+ > + k π, for some positive integer k. Then nd hence lso s k ( k + m increses. ( k m+ k / Y ν (kπ pν (kπ ( k m k / J ν (kπ pν (kπ increses to /π, decreses to /π, ( k m k / J ν (kπ decreses to /π, The proof is similr to tht of Corollry 5.0.

11 MONOTONIC SEQUENCES RELATED TO ZEROS OF BESSEL FUNCTIONS References [] S. Akiym nd Y. Tnigw, Slem numbers nd uniform distribution modulo, Publ. Mth. Debrecen 64 (004, [] Á. Elbert, An pproximtion for the zeros of Bessel functions, Numer. Mth. 59 (99, [3] Á. Elbert, Some recent results on the zeros of Bessel functions nd orthogonl polynomils, Proceedings of the Fifth Interntionl Symposium on Orthogonl Polynomils, Specil Functions nd their Applictions (Ptrs, 999, J. Comput. Appl. Mth. 33 (00, no. -, [4] Á. Elbert, L. Gtteschi nd A. Lforgi, On the concvity of zeros of Bessel functions, Appl. Anl. 6 (983, [5] Á. Elbert nd A. Lforgi, On the squre of the zeros of Bessel functions, SIAM J. Mth. Anl. 5 (984, 06. [6] Á. Elbert nd A. Lforgi, Monotonicity properties of the zeros of Bessel functions, SIAM J. Mth. Anl. 7 (986, [7] Á. Elbert nd A. Lforgi, Further results on McMhon s symptotic pproximtions, J. Phys. A: Mth. Gen. 33 (000, [8] L. Gtteschi, Vlutzione dell errore nell formul di McMhon per gli zeri dell J n(x di Bessel nel cso 0 n, Rivist Mt. Univ. Prm (950, [9] L. Gtteschi, Funzioni Specili, UTET, Torino, 973. [0] L. Gtteschi, Asymptotics nd bounds for the zeros of Lguerre polynomils: survey, J. Comput. Appl. Mth. 44 (00, 7 7. [] P. Hrtmn, Ordinry Differentil Equtions, Wiley, New York, 964. [] A. Lforgi, Sugli zeri delle funzioni di Bessel, Clcolo 7 (980, 0. [3] L. Lorch nd P. Szego, Higher monotonicity properties of certin Sturm-Liouville functions, Act Mth. 09 (963, [4] M. E. Muldoon, Continuous rnking of zeros of specil functions, J. Mth. Anl. Appl., to pper. [5] G. N. Wtson, A Tretise on the Theory of Bessel Functions, nd ed., Cmbridge University Press, 944. Deprtment of Mthemtics & Sttistics, York University, Toronto, Ontrio M3J P3, Cnd E-mil ddress: lorch@mthstt.yorku.c, muldoon@yorku.c