COMPUTATION OF SOME WONDERFUL RESULTS INVOLVING CERTAIN POLYNOMIALS

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Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: 95-0056 Volume: 0 Issue: 0 Apr-05 www.irjet.

1 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: COMPUTATION OF SOME WONDERFUL RESULTS INVOLVING CERTAIN POLYNOMIALS Salahuddi, Itazar Husai Assistat Professor, Applied Scieces, P.D.M College of egieerig, Haryaa, Idia Assistat Professor, Applied Scieces, D. I. T. M. R, Haryaa, Idia *** Abstract - I this paper we have established some The series coverges for all fiite z if A B, coverges idefiite itegrals ivolvig certai polyomials i for z < if A=B+, diverges for all z, z 0 if A > B+. the form of Hypergeometric fuctio.the results Lucas Polyomials represet here are assume to be ew. Key Words: Hypergeometric fuctio, Lucus Polyomials, Gegebaur Polyomials, Hermoic umber, Beroulli Polyomials, ad Hermite Polyomials.. Itroductio The special fuctio is oe of the cetral braches of Mathematical scieces iitiated by L Euler.But systematic study of the Hypergeometric fuctios were iitiated by C.F Gauss, a immiet Germa Mathematicia i 8 by defiig the Hypergeometric series ad he had also proposed otatio for Hypergeometric fuctios. Sice about 50 years several taleted brais ad promisig Scholars have bee cotributed to this area. Some of them are C.F Gauss, G.H Hardy, S. Ramauja,A.P Prudikov, W.W Bell, Yu. A Brychkov ad G.E Adrews. We have the geeralized Gaussia hypergeometric fuctio of oe variable AF B(a,a,,a A;b,b, b B;z ) = () The Lucas polyomials are the w-polyomials obtaied by settig p() = ad q() = i the Lucas polyomials sequece. It is give eplicitly by L ) + ( 4 ) ] () ( ) [( 4 The first few are L L ( ) () L ( ) where the parameters b, b,.,b B are either zero or egative itegers ad A, B are o egative itegers. The series coverges for all fiite z if A B, coverges for z < if A=B+, diverges for all z, z 0 if A > B+. L ( ) Geeralized Harmoic Number 05, IRJET.NET- All Rights Reserved Page 8

0 (6) I the limit of the geeralized harmoic umber coverges to the Riema zeta fuctio ( m) lim H ( m) Beroulli Polyomial (5) I Mathematics, Gegebauer polyomials or ultraspherical polyomials are

2 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: The geeralized harmoic umber of order of m is give by H ( m) k k m (4) Gegebauer polyomials t te t B t e! 0 (6) I the limit of the geeralized harmoic umber coverges to the Riema zeta fuctio ( m) lim H ( m) Beroulli Polyomial (5) I Mathematics, Gegebauer polyomials or ultraspherical polyomials are orthogoal polyomials o the iterval [-,] with respect to the weight fuctio ( ).They geeralize Legedre polyomials ad Chebyshev polyomials, ad are special cases of Jacobi polyomials. They are amed after Leopold Gegebauer. Eplicitly, ( ) k ( k) ( k) ( ) ( ) ( ) k0 ( ) k!( k)! C z (7) Laguerre polyomials I mathematics, the Beroulli polyomials occur i the study of may special fuctios ad i particular the Riema zeta fuctio ad Hurwitz zeta fuctio. This is i large part because they are a Appell sequece, i.e. a Sheffer sequece for the ordiary derivative operator. Ulike orthogoal polyomials, the Beroulli polyomials are remarkable i that the umber of crossig of the -ais i the uit iterval does ot go up as the degree of the polyomials goes up. I the limit of large degree, the Beroulli polyomials, appropriately scaled, approach the sie ad cosie fuctios. Eplicit formula of Beroulli polyomials is! B b k k! k! k, for 0, where b k are the k0 Beroulli umbers. The Laguerre polyomials are solutios to the Laguerre differetial equatio y '' ( ) y' y 0, which is a special case of the more geeral associated Laguerre differetial equatio, defied by The geeratig fuctio for the Beroulli polyomials is 05, IRJET.NET- All Rights Reserved Page 8

d The Laguerre polyomials are give by the sum k

The Hermite polyomials H ( ) cotour itegral!

Legedre fuctios of the first kid, Legedre coefficiets, or zoal harmoics (Whittaker ad Watso 990, p. 0), are solutios to the Legedre differetial equatio.

3 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: y '' ( ) y' y 0, where ad are real umbers with =0. Legedre fuctio of the first kid The Laguerre polyomials are give by the sum k ( )! L k! k! k! k0 k (8) Hermite polyomials The Hermite polyomials H ( ) are set of orthogoal polyomials over the domai (-, ) with weightig fuctio e. The Hermite polyomials H ( ) cotour itegral! r rz H() z e t dt i, ca be defied by the Where the cotour icloses the origi ad is traversed i a couterclockwise directio The Legedre polyomials, sometimes called Legedre fuctios of the first kid, Legedre coefficiets, or zoal harmoics (Whittaker ad Watso 990, p. 0), are solutios to the Legedre differetial equatio. If l is a iteger, they are polyomials. The Legedre polyomials P ( ) are illustrated above for ad =,,..., 5. The Legedre polyomials P ( ) ca be defied by the cotour itegral P ( z) ( tz t ) t dt i, (0) where the cotour ecloses the origi ad is traversed i a couterclockwise directio (Arfke 985, p. 46). Legedre fuctio of the secod kid (Arfke 985, p. 46). The first few Hermite polyomials are H 0 H H 4 (9) H ( ) 8 H 4 4 ( ) 6 48 The secod solutio to the Legedre differetial equatio. The Legedre fuctios of the secod kid 05, IRJET.NET- All Rights Reserved Page 84

The first few are Q0 l( ( ) Q l( ) Q l( 4 )- )- () T 0 T T ( ) T ( ) 4 A beautiful plot ca be obtaied by plottig radially, icreasig the radius for each

Chebyshev polyomial of the first kid The Chebyshev polyomials of the first kid are a set of orthogoal polyomials defied as the solutios to the Chebyshev

They are also itimately coected with trigoometric multiple-agle formulas.

4 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: satisfy the same recurrece relatio as the Legedre polyomials. The first few are Q0 l( ( ) Q l( ) Q l( 4 )- )- () T 0 T T ( ) T ( ) 4 A beautiful plot ca be obtaied by plottig radially, icreasig the radius for each value of, ad fillig i the areas betwee the curves (Trott 999, pp. 0 ad 84). Chebyshev polyomial of the first kid The Chebyshev polyomials of the first kid are a set of orthogoal polyomials defied as the solutios to the Chebyshev differetial equatio ad deoted. They are used as a approimatio to a least squares fit, ad are a special case of the Gegebauer polyomial with. They are also itimately coected with trigoometric multiple-agle formulas. The Chebyshev polyomials of the first kid are defied through the idetity T (cos θ)=cos θ. Chebyshev polyomial of the secod kid The Chebyshev polyomial of the first kid defied by the cotour itegral ca be ( t ) t T () z 4i tz t dt, () where the cotour ecloses the origi ad is traversed i a couterclockwise directio (Arfke 985, p. 46). The first few Chebyshev polyomials of the first kid are 05, IRJET.NET- All Rights Reserved Page 85

They are a special case of the Gegebauer polyomial with. They are also itimately coected with trigoometric multiple-agle formulas.

t g( t) ( e ), givig the geeratig fuctio t e t E t. (4) e!

5 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: E 0 A modified set of Chebyshev polyomials defied by a slightly differet geeratig fuctio. They arise i the developmet of four-dimesioal spherical harmoics i agular mometum theory. They are a special case of the Gegebauer polyomial with. They are also itimately coected with trigoometric multiple-agle formulas. The first few Chebyshev polyomials of the secod kid are U 0 E E ( ) (5) E 4 Geeralized Riema zeta fuctio U () U ( ) 4 U ( ) 8 4 Euler polyomial The Euler polyomial sequece with is give by the Appell t g( t) ( e ), givig the geeratig fuctio t e t E t. (4) e! 0 The first few Euler polyomials are The Riema zeta fuctio is a etremely importat special fuctio of mathematics ad physics that arises i defiite itegratio ad is itimately related with very deep results surroudig the prime umber theorem. While may of the properties of this fuctio have bee ivestigated, there remai importat fudametal cojectures (most otably the Riema hypothesis) that remai uproved to this day. The Riema zeta fuctio is defied over the comple 05, IRJET.NET- All Rights Reserved Page 86

6 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: plae for oe comple variable, which is covetioally deoted s (istead of the usual ) i deferece to the otatio used by Riema i his 859 paper that fouded the study of this fuctio (Riema 859). for is ot a coicidece sice it turs out that mootoic decrease implies the Riema hypothesis (Zvegrowski ad Saidak 00; Borwei ad Bailey 00, pp ). O the real lie with, the Riema zeta fuctio ca be defied by the itegral u du, where ( ) 0 u e fuctio. Comple ifiity is the gamma The plot above shows the "ridges" of for ad. The fact that the ridges appear to decrease mootoically Comple ifiity is a ifiite umber i the comple plae whose comple argumet is ukow or udefied. Comple ifiity may be retured by Mathematica, where it is represeted symbolically by CompleIfiity. The Wolfram Fuctios Site uses the otatio comple ifiity. to represet. MAIN RESULTS ( )( ) ( ) ( ) L e d e e [ { F (, ; ; e )} { F(,, ;, ; e )}] C (7) Here L () is Lucas Polyomials. ( ) H si d cossi si F (, ( ); ;cos ) C (8) Here H is Geeralized Harmoic umber. si d C (9) Here ζ() is Geeralized Riema zeta fuctio. 05, IRJET.NET- All Rights Reserved Page 87

7 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: cos{ F(, ; ( );si )} H si d si [ ( )si{ F (,, ; ( ), ;si )}] C (0) Here H () is Hermite polyomials. cos{ F(, ; ( );si )} U si d si [ ( )si{ F (,, ; ( ), ;si )}] C () Here U () is Chebyshev polyomial of the secod kid. T si d si [ cos{ F(, ; ( );si )} ( )si{ F (,, ; ( ), ;si )}] C () Here T () is Chebyshev polyomial of the first kid. P si d si [ cos{ F(, ; ( );si )} ( )si{ F (,, ; ( ), ;si )}] C () Here P () is Legedre fuctio of the first kid. Hsec d sicos sec { F (, ( ); ;si )} C (4) ( ) H cos ec d cossi cos ec F (, ( ); ;cos ) C (5) 05, IRJET.NET- All Rights Reserved Page 88

8 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: F si d si [ cos{ F(, ; ( );si )} ( )si{ F (,, ; ( ), ;si )}] C L si d si [ cos{ F(, ; ( );si )} ( )si{ F (,, ; ( ), ;si )}] C cos{ F(, ; ( );si )} C si d si [ ( )si{ F (,, ; ( ), ;si )}] C (6) (7) (8) Here C () is Gegebauer polyomials. B cos d cos [ ( ) ( )cos si{ F(, ; ( );cos )} * F(,, ; ( ), ;cos ) sicos { F (, ( ); ( );cos C ( ) si (9) Here B () is Beroulli Polyomial. sicos F(, ( ); ( );cos. cos ( ) H d C (0) ( ) si 05, IRJET.NET- All Rights Reserved Page 89

9 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: L si d si [ ( )si * F(,, ; ( ), ;si ) cossi F (, ; ;cos ) cos F(, ; ( );si ) ] C () Here L () is Laguerre polyomials. B cos ec d cos ec [ ( ) ( )si * F(,, ; ( ), ;si ) ( ) cossi F(, ( ); ;cos ) cos F(, ; ( );si ) ] C () REFERENCES [] A. P. Prudikov, O. I. Marichev,, ad Yu. A. Brychkov, Itegrals ad Series, Vol. : More Special Fuctios., NJ:Gordo ad Breach, 990. [] G. Arfke, Mathematical Methods for Physicists, Orlado, FL: Academic Press. [] L.C. Adrews, Special Fuctio of mathematics for Egieers,Secod Editio,McGraw-Hill Co Ic., New York, 99. [4] Milto Abramowitz, Iree A. Stegu, Hadbook of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables, New York: Dover, 965. [5]. Richard Bells,, Roderick Wog, Special Fuctios, A Graduate Tet, Cambridge Studies i Advaced Mathematics, 00. [6] T. J. Rivli, Chebyshev Polyomials., New York: Wiley, 990. [7] Wikipedia(0,October), Beroulli Polyomials. Retrived from website:http: //e.wikipedia.org/wiki /Beroulli_polyomials. [8] Wikipedia(0,October), Gegebaur Polyomials. Retrived from website: http: //e.wikipedia.org /wiki /Gegebauer_polyomials. [9] WolframMathworld(0,October),Riema Zeta fuctio. Retrived from website: http: //mathworld.wolfram.com /RiemaZetaFuctio.html. [0] WolframMathworld(0,October), Harmoic Number. Retrived from website: http: //mathworld.wolfram.com/harmoicnumber.html. []WolframMathworld(0,October), LaguerrePolyomials. Retrived from website: [] WolframMathworld(0,October), Hermite Polyomials. Retrived from website: 05, IRJET.NET- All Rights Reserved Page 90

10 Iteratioal Research Joural of Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 0 Apr-05 p-issn: []WolframMathworld(0,October),Legedre Polyomial. Retrived from website: [5] WolframMathworld(0,October), Lucas polyomials. Retrived from website: [4]WolframMathworld(0,October), Chebyshev Polyomial of the first kid. Retrived from website: hefirstkid.html. 05, IRJET.NET- All Rights Reserved Page 9

Introductions to HarmonicNumber2

Introductions to HarmonicNumber2 Itroductios to HarmoicNumber2 Itroductio to the differetiated gamma fuctios Geeral Almost simultaeously with the developmet of the mathematical theory of factorials, biomials, ad gamma fuctios i the 8th