ON RUEHR S IDENTITIES
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1 ON RUEHR S IDENTITIES HORST ALZER AND HELMUT PRODINGER Abstract We apply completely elemetary tools to achieve recursio formulas for four polyomials with biomial coefficiets I particular, we obtai simple ew proofs for Ruehr s combiatorial idetities Moreover, we use our formulas to fid idetities ad iequalities for trigoometric polyomials Mathematics Subject Classificatios: 05A19, 11C08, 26D05 Keywords: Combiatorial idetities, recursio formula, algebraic ad trigoometric polyomials, iequalities 1
2 2 H ALZER AND H PRODINGER We defie the followig four sums: ( 3 j A 3 j 2 C 2 1 Itroductio ad mai result ( 3 j ( 3 j, B, D 2 2 j, j ( 4 j ( j The study of two itegral equatios led Ruehr [6] to the idetities (1 A C ad B D ( 0, 1, 2, We remark that i [6] A is erroeously give with 4 j istead of 3 j The corrected versio is due to Meeha et al [8] I a recetly published paper, Meeha et al [8] preset ew computer-geerated proofs for (1 by usig the Wilf-Zeilberger method Moreover, they offer a iterestig combiatorial proof for A B ( 0, 1, 2, I particular, the authors show that A, B, C, ad D satisfy the recursio formula (2 X 0 1, X ( 4 X ( 0, 1, 2, 4( + 1 I this ote, we establish the recursios i the simplest possible way, by oly usig the recursio ( ( ( k + 1 k + 1 k of Pascal s triagle ad elemetary rearragemets Actually, we prove a bit more: i the et sectio we demostrate that the four polyomials ( 3 j A ( j, B ( j, 2 j C ( 2 ( 3 j satisfy the followig recursio formulas j, D ( 2 j j Theorem For all 0 we have 3 A +1 ( ( 1 A ( + (4 + ( 22 ( , 2 2( + 1( 1 2 ( + 13 B +1 ( B 2 ( + (4 + ( 22 ( , 2( + 1 2
3 ON RUEHR S IDENTITIES 3 C +1 ( 3 1 C ( + (2 + ( 22 + ( ( + 1( 1 ( + 13 D +1 ( D ( + ( ( ( + 1 Remark 1 From the recursios we obtai the idetities (3 A ( + 1 B ( ad C ( + 1 D ( The fact that A ( + 1 B ( ca be see directly: A ( + 1 ( 3 j ( + 1 j 2 ( ( j 3 j k k 2 jk k B ( k k0 k0 k0 k0, ( j ( ( j 3 j k k 2 k k The idetity that was used here is a variat of the Vadermode covolutio [5] The direct proof that C ( + 1 D ( is similar Remark 2 We have A (3 A, B (2 B, C ( 3 C, ad D ( 4 D From the Theorem we coclude that A, B, C ad D satisfy the recursio formula (2 I particular, we obtai A B C D for 0 I the et sectio, we prove our theorem ad i Sectio 3 we show that (3 ca be applied to obtai idetities ad iequalities for trigoometric polyomials Let us start with the simpler oes +1 ( B +1 ( j + 1 j j 1 2 Proof +1 [( ( j j j ( j + 3 j j j + 3 j j j2 ( j j j + ( ] j
4 4 H ALZER AND H PRODINGER D +1 ( 1 ( + 13 j 2 j ( + 13 B 2 ( + ( ( ( ( j j [( ( j j j j j ( j + + j j j j j j 1 + j j j + + j 1 ( ( ] j j + 2 j j j j 1 j 2 ( + 13 D ( 1 ( ( ( ( ( + 13 D ( + (2 + ( 22 + ( ( + 1 Now we move to the two remaiig sums +1 ( j A +1 ( +1 j j ( j j + j ( j j + j ( j +1 j j 1 ( j +1 j + 1 j A +1( 1 ( j j 1 j ( (
5 Therefore ( 1 1 A +1 ( + 1 Simplifyig, ( ( ON RUEHR S IDENTITIES 5 ( 2 + j j + j ( j +1 j j ( 2 + j +1 j j 1 +1 ( j A ( + + j j A ( + 1 ( A +1 ( A ( + Therefore Fially, A +1 ( C +1 ( ( [ ( 1 1 A +1 ( + 1 ( ( 1 A ( + ( ( ( + 1( ( j 2+2 j j ( ] ( ( ( ( j ( + j 2+2 j j j j 1 ( ( C ( ( j j 1 ( j ( ( C ( C +1( 1 ( or C +1 ( 3 1 C ( + ( ( ( + 1( 1 ( 3 + 1
6 6 H ALZER AND H PRODINGER 3 Applicatios Gould [4] collected umerous iterestig biomial idetities for trigoometric sums ad polyomials We use the formulas give i (3 to obtai idetities for sie ad cosie sums which we could ot locate i Gould s compilatio or ay other publicatio We set e iθ The, ad A ( + 1 ( 3 j (e iθ + 1 j 2 B ( e iθj j Sice A ( + 1 B (, we obtai (4 ad (5 ν0 ( j ( 3 j j [cos(νθ ] + i si(νθ 2 ν ( j ( 3 j j cos(νθ 2 ν ( j ( 3 j j si(νθ 2 ν ν1 ν0 [cos(jθ ] + i si(jθ j cos(jθ j si(jθ j If we apply C ( + 1 D (, the we fid the followig compaios of (4 ad (5: ad 2 2 ( j ( 3 j j cos(νθ ν ν0 ( j ( 3 j j si(νθ ν ν1 2 2 cos(jθ j si(jθ j A theorem of Vietoris (see [7] ad [10] states that if a 0, a 1,, a are real umbers satisfyig (6 a 0 a 1 a > 0 ad a 2j 2j 1 a 2j 1 2j the a j cos(jθ ad a j si(jθ > 0 (1 j /2, (0 < θ < π A short calculatio reveals, that if we set a j ( 3+1 j (j 0, 1,,, the (6 is valid If follows that the sums give i (4 ad (5 are positive for all N ad θ (0, π
7 ON RUEHR S IDENTITIES 7 If we replace i (5 θ by π θ, ad add up the two sums o both sides, the we arrive at ( j ( 3 j j (7 si(νθ si(jθ > 0 ( N; 0 < θ < π 2 ν j ν1 ν odd j odd We obtai similar results if we apply A ( + 1 B ( with e iθ 1 I particular, we get the followig couterpart of (7: (8 ( j ( j ( 3 j ( 1 j 1 si(νθ si(jθ > 0 ( N; 0 < θ < π j ν 2 (9 2 ν1 ν odd Aother relative of (7 is give by ( 1 j 1 ( j j ν1 ν odd j odd ( j si(νθ ν 2 j odd ( 3 j si(jθ > 0 ( N; 0 < θ < π We set e iθ 1 i C ( + 1 D ( The (as before, we replace θ by π θ ad add up This yields that the two sie polyomials i (9 are equal To prove the positivity we make use of the kow idetity (10 b ν si((2ν 1θ (b k b k+1 si2 (kθ (b +1 0 si(θ ν1 k1 Let b k ( 3+1 k (k 1, 2,, ad b+1 0 The, b k > b k+1 (k 1,,, so that (10 ad 2 j odd ( 3 j si(jθ b ν si((2ν 1θ reveal that the sums i (9 are positive for N ad θ (0, π We ote that the costat lower boud 0, give i (7, (8 ad (9, respectively, is best possible Additioal iequalities for trigoometric polyomials ivolvig biomial coefficiets are give i the research papers [1], [2], [3] Noegative trigoometric polyomials have remarkable applicatios i various braches For istace, they play a importat role i geometric fuctio theory, approimatio theory ad i the theory of absolutely mootoic fuctios More iformatio o this subject ca be foud i [9, chapter 4] ν1 Ackowledgemet We are grateful to the referee for ispirig commets which improved the quality of the paper
8 8 H ALZER AND H PRODINGER Refereces [1] H Alzer ad B Fuglede, O a trigoometric iequality of Turá, J Appro Th 164 (2012, [2] H Alzer ad S Koumados, Remarks o a sie polyomial, Arch Math 93 (2009, [3] H Alzer ad S Koumados, Sharp estimates for various trigoometric sums, Aalysis 33 (2012, 9 26 [4] H W Gould, Combiatorial Idetities: Table III: Biomial idetities derived from trigoometric ad epoetial series, 26 pages, wwwmathwvuedu/ gould/ [5] R L Graham, D E Kuth, ad O Patashik, Cocrete Mathematics (Secod Editio, Addiso Wesley, 1994 [6] N Kimura ad O G Ruehr, Chage of variable formula for defiite itegrals, E 2765, Amer Math Mothly 87 (1980, [7] S Koumados, A etesio of Vietoris s iequalities, Ramauja J 14 (2007, 1 38 [8] S Meeha, A Tefera, M Weselcouch, ad A Zeleke, Proofs of Ruehr s idetities, INTEGERS 14 (2014, paper A10 [9] G V Milovaović, D S Mitriović, ad Th M Rassias, Topics i Polyomials: Etremal Problems, Iequalities, Zeros, World Sci, Sigapore, 1994 [10] L Vietoris, Über das Vorzeiche gewisser trigoometrischer Summe, Sitzugsber Öst Akad Wiss 167 (1958, ; Az Öst Akad Wiss 1959, Horst Alzer, Morsbacher Str 10, Waldbröl, Germay address: HAlzer@gmde Helmut Prodiger, Mathematics Departmet, Stellebosch Uiversity, 7602 Stellebosch, South Africa address: hprodig@suacza
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