Hypergeometric Functions and Lucas Numbers

Transcription

1 IOSR Jourl of Mthetis (IOSR-JM) ISSN: Volue Issue (Sep-Ot. ) PP - Hypergeoetri utios d us Nuers P. Rjhow At Kur Bor Deprtet of Mthetis Guhti Uiversity Guwhti-78Idi Astrt: The i purpose of this pper is to give the reltioship etwee hypergeoetri series d the us sequee. A vriety of represettios i ters of fiite sus d ifiite series ivolvig oeffiiets re otied. While y of the re well ow d soe idetities pper to e ow. I. Hypergeoetri utios: The hypergeoetri futio ( x) is defied y the series (.)! or x < d y otiuous elsewhere (see[] p.) where the risig ftoril is defied y =(+)(+) (+-) () (.) II. us Nuers We defie the th us uer deoted y s d (.) Agi we hve Settig = = we rrive t (.) with (.)the represettio of (. )eoes (.) whih is well ow ([].p.8) Throughout this hpter the followig results will e used (.)!! (.) (.) P g e

2 !! Hypergeoetri utios Ad us Nuers (.7) ( x) si x (.)!! (.) III. ier Ad Qudrti Trsfortio I this setio we will use the well-ow lier d qudrti trsfortios for the hypergeoetir futios to derive soe represettio fro (.). We egi y the followig pir of lier trsfortios: (.) (.) Tht re lied together y the reltio (.) ([].p..theore ) We lso ote the ovious reltioship Applyig (.) to (.) the RHS(.) of is fiite su oly whe is odd d we get the followig idetity (.) A opio reltioship of (.) e otied y pplyig (.) to (.). I this se hs to e eve d (.) (.8) Our ext trsfortio forul is P g e

3 Hypergeoetri utios Ad us Nuers P g e (.) However sie +-=- i(.) oe of the g ters i uertor is ot defied. So we use the followig trsfortio whih is speil se where or is egtive iteger d is o egtive iteger. (.7) (see[8].p.) Also we use (see[].p.) The we hve (.8) Aother trsfortio forul is (.) We pply this to (.) d usig the ft tht () hs poles t o-positive itegers we see tht oe of two ters i (.) lwys disppers. Evlutig the g ters i the reiig expressio we rrive t (.) Ad

4 Hypergeoetri utios Ad us Nuers P g e (.) ro (.) we hve (.) Ad (.) These two foruls give us the first ifiite series represettio for the us Nuers. Eployig (.) d (.) to (.7) we rrive t (.) (.) The ext trsfortio forul is ) ( () (.) Applyig (.) to (.) we get (.7) (.8) Applyig (.) to these we get

5 Hypergeoetri utios Ad us Nuers P g e (.) (.) Now we use qudrti trsfortios. Our first qudrti trsfortio is (.) (see[].p.8.(.)) Eployig this to (.) d (.) we get the followig trsfortios respetively (.) Ad (.) Agi pplyig (.) we get (.) Ad (.) Eployig (.) to (.) d (.7) to (.) we get (.)

6 Hypergeoetri utios Ad us Nuers P g e Ad (.7) ro (.) d (.) we get. (.8) Agi pplyig(.) to (.) we get (.) or odd we hve fro (.7)y usig (.) tht! (.) Agi for eve eployig (.) to (.7) we rrive t! (.) Agi the idetity (.) e further trsfored y forul (.) s follows (.) Applyig (.) to idetity (.) we get

7 Hypergeoetri utios Ad us Nuers 7 P g e (.) Ad (.) Agi pplyig (.) to these two idetities we rrive t (.) (.) illy we use the followig two qudrti trsfortio forule (.7) (.8) Applyig these two to (.) we get (.) (.) IV. Expliit oruls I this setio we will siply rewrite the foruls otied ove i ters of oitoril sus. Idetities (.8) (.7) d (.8) respetively led to the su (.)

8 Hypergeoetri utios Ad us Nuers 8 P g e (.) (.) Agi Idetities (.) (.) d (.) give rise to (.) (.) (.) Both (.) d (.) led to the followig idetity (.7) Agi oth (.) d (.) led to the followig (.8) Agi fro (.) d (.7) we get respetively (.) (.) ro the idetities (.) (.)(.) d (.) we respetively rrive t (.) (.) (.)

9 Hypergeoetri utios Ad us Nuers P g e (.) Usig (.)(.) d (.) respetively we rrive t (.) (.) (.7) V. urther Applitio The Geerlied Hypergeoetir utio is defied y! q i j p i i p p q p (.) (see[].p.7) We use the followig idetity due to Cluset (see[8].p.8) (.) Tig = =+ the (.7) d (.) gives (.) By wy of (.) we get (.) Referees [] A.K.Agrwl ``O ew id of uers" The ioi Qurterly Vol 8() ( ) -. [] G.E. Adrews Rihrd Asey Rj Roy ``Speil utios"cridge Uiversity press. [] W.N. Biley.`` Geerlied Hypergeoetri Series". Crige: Cridge Uiversity Press. [] C.Jord. ``Clulus of iite Differees". New Yor : Chelse. [] Thos Koshy. ``ioi d us Nuers with Applitios". [] E.D. Riville. ``Speil utios" New Yor : Mill 7. [7] J.Riord. ``Citoril Idetities". Hutigto. New Yor: Krieger 7. [8] M.Arowit d I.A.Stegu. ``Hdoo of Mthetil utios". Wshigto D.C.: Ntiol Bureu of stdrds.