SOME INTRIGUING LIMITS CONTINUATION

HTML
DOWNLOAD

Wielkość: px

Rozpocząć pokaz od strony:

Download "SOME INTRIGUING LIMITS CONTINUATION"

Weronika Łuczak
5 lat temu
Przeglądów:

1 ZESZYTY NAUKOWE POLITECHNIKI ŚLĄSKIEJ 2013 Seria: MATEMATYKA STOSOWANA z. 3 Nr kol Piotr LORENC, Roman WITUŁA Institute of Mathematics Silesian University of Technology SOME INTRIGUING LIMITS CONTINUATION Summary. Wituła and Słota in[college Math. J , 328] proposedawayofprovingtherelation1givenbelowwhichappearedtobe agenuineresult.authorsofthepresentpaper,inspiredbytheformofthis it,trytofindsomegeneralizationsofthisone,alsointhecontetof some special functionse.g. the gamma function, the generalized Laguerre polynomials. O PEWNYCH INTRYGUJĄCYCH GRANICACH KONTYNUACJA Streszczenie. Wituła i Słota w notce[college Math. J , 328] zaproponowali udowodnienie relacji1, podanej poniżej, która wydaje się bardzo ciekawą zależnością. Autorzy niniejszego artykułu, zainspirowani postacią tej granicy, próbują znaleźć różne jej uogólnienia także w kontekście pewnych funkcji specjalnychnp. funkcji gamma, uogólnionych wielomianów Laguerre a Mathematics Subject Classification: 40A05. Corresponding author: R. WitułaRoman.Witula@polsl.pl. Received: r. Piotr Lorenc will be net year the MSc graduate student in Mathematics.

2 18 P. Lorenc, R. Wituła Problem of evaluating the its of functions depending on the epressions 1+ α islikeneverendingstory.lookatthefollowingonessee[15]: foreveryn N; e 1n n... e 1 4 e 1 3 e 1 2 e e 1n n+1 1 }{{} n+1 times epa n... epa 2 epa 1 r i1 r j1 1+ α i βi γ 1+ j δj... }{{} n times foreveryn,r N,α i,β i,γ i,δ i R,i1,...,rwhere A k : 1 α k δ k γ k β :1 k α k β γ k δ, k N, ep A n+1, 2 andthesymbol denotesthescalarproductappliedtothevectors α k, β, γ k, δ R r, defined in the following way α k :[ α 1 k, α 2 k,..., α r k ], β:[β 1,β 2,...,β r ], γ k :[ γ 1 k, γ 2 k,..., γ r k ], δ:[δ 1,δ 2,...,δ r ]; 2αβ 1+ 1 α 1+ 1 β ep 11 1 α βe α β 12 α +1 3 β foreveryα,β R\{0},α β; ln ln ep 5 ; 4 12 e 1+ 1 e 2 ; 5 2 e e e. 6 24

3 Some intriguing its continuation 19 Sketchoftheproofsof1and3. Wehave>1: 1+ 1 ep ln 1+ 1 ep n0 whichimplies1.now,letα,β R\{0},α β.thenweget 1+ 1 α ep which implies α 1+ 1 β βep αln 1+ 1 α 1 1 2α + 1 3α 2 2+o 1 2 ep [ 1 ep eep ep 1 1 2β + 1 3β 2 2+o n n+1 n ep βln 1+ 1 β 1 1 2β + 1 3β 2 2+o 1 2 2β 1 2α α 2 2 3β 2 2+o β +o ] 1 [ α β 2αβ α β5α+11β 24α 2 β 2 2 +o 1 ] 2, 2αβ [ 1+ 1 α 1+ 1 β ] α βe α β ep 1 2β +o1 1 5α+11β 1, 12αβ +o which easily proves3. Proof of2. Wehave>ma{ α, β }: 1+ α β 1+ γ δ ep which implies2. Sketchoftheproofsof5and6. Bysubstitution 1 tweget βln 1+ α δln 1+ γ ep 1 n βα n+1 δγ n+1 1 n0 e e 1+t t, t 0 + t n+1 n,

4 20 P. Lorenc, R. Wituła 2 e e e 2 t 1+t e 1 t t 0 + t 2. Moreoverwehave>1 t 0,1: 1+t 1 1 t ep ep t ln1+t n0 1 n n+1 tn. 7 Application of l Hospital s rule and relation7 gives the values of both its. Proof of4. We have and which implies4, since ln o 2 ln o o, ep 12 +o Remark1.Wenotethatit1forn1is aregularguest ofmanyproblems incalculusbooksseee.g[4,6,8].incontrast,theitofgeneralshape1and its generalizations2 are probably new. Remark2.Limits3 6,whichareprobablynewseee.g.[4,6 8,10,11,15], arose during the discussion on generalizations of its1 and2and they are far from our epectations. Authors hope that these its inspire the Readers to look for another results. Remark3.Wenotethat αe + α foreveryα R.

5 Some intriguing its continuation 21 Proof. From the following formula 1+ a ep ln 1+ a epaep e a k0 1 k! n1 e a 1 a2 2 + a n a n+1 n+1 n 1 n a n+1 n+1 n k n1 a 3 a 2 a a fora1andfromthefollowingdecompositionwhichisaspecialformofbinomial series: 1+ 1 α1+ α 1 α α +o α weget α 1+ 1 α α 1+ 1 α +e 1+ 1 α α e + e α 1, e 2 +e 2 +αeαe. Remark4.Inpaper[5]theauthorsobservedthatfromformula1forn1 and from Stirling s formulasee e.g.[9] the following approimation holds 2π 2 + Γ+1. e +1 2 StartingfromthisformulaFengandWangauthorsof[5]provedthatforsufficientlylarge Rthefollowingoneholds Γ+1 2π whereconstantsα k satisfytherecurrencerelation k j0 m 2 + α k 2k+O 1 2m+2 k0 α j 2k 2j k+3 B 2k+2 22k+12k+2 fork0,1,2,...andb n denotesthen-thbernoullinumberseee.g.[9].,

6 22 P. Lorenc, R. Wituła Remark 5. In paper[13] the asymptotic epansion of the Gamma functions ratio is obtained Γz+α Γz+β zα β 1+α βα+β 1 1 2z + α β 1 + 3α+β 1 2 α+β z 2+O z 3 whereα,β,z Candz.Fromthisepansionwededucetherelation z z β αγz+α 1 z Γz+β 2 α βα+β 1, where each power has its principal value., Remark6.Inpaper[14]theformulaeoftype1connectedwithit n n n! n e 1 are discussed. For eample, there are presented the following relations n n e n! n n 1 2πn n e 1 12 and n n! 2π e n 2 +n+ 1 6 n+ 1 2 n 3 e Remark7.LetL a n z denote the n-th generalized Laguerre polynomialsee [12]. Then we obtain the relationsee[1 3]: e L a z 2 n n z 2 e 2 nz 1 n π z 1 4 a 2n 1 4 +a a az+4z 2 z forz C\,0],a C,whereRe nzdenotes θ n z cos 2 andθ: argz π,π].

7 Some intriguing its continuation 23 References 1. Assche van W.: Weighted zero distribution for polynomials orthogonal on an infinite interval. SIAM J. Math. Anal , Assche van W.: Erratum to Weighted zero distribution for polynomials orthogonal on an infinite interval. SIAM J. Math. Anal , Borwein D., Borwein J.M., Crandall R.E.: Effective Laguerre asymptotics. SIAM J. Numer. Anal , Dorogovcev A.J.: Mathematical Analysis Problems. Wisza Szkola, Kiev 1987 in Russian. 5. Feng L., Wang W.: Two families of approimations for the gamma function. Numer. Algorithms , Kudravcev L.D., Kutasov A.D., Czechlov W.I., Szabunin M.I.: Problems in Mathematical AnalysisLimits, Continuity, Differentiability. Phys.-Mathem. Literature Press, Moscov 1984in Russian. 7. Lorenc P., Wituła R.: Limits and functions connected with ein preparation. 8. Polya G., Szegö G.: Aufgaben und Lehrsätze aus der Analysis. Erster Band, Springer, Berlin RabsztynSz.,SłotaD.,WitułaR.:GammaandBetaFunctions,PartI.Wyd. Pol. Śl., Gliwice 2012in Polish. 10. Sadovniczij W.A., Grigorian A.A., Koniagin S.W.: Problems for Students Mathematical Olympiads. Moscov University Press, Moscov 1987in Russian. 11. Sadovniczij W.A., Podkolzin A.S.: Problems for Students Mathematical Olympiads. Nauka Press, Moscov 1978in Russian. 12. Szegö G.: Orthogonal Polynomials. American Mathematical Society, Providence Tricomi F., Erdelyi A.: The asymptotic epansions of a ratio of gamma functions. Pacific J. Math , WitułaR.,JamaD.,NowakI.,OlczykP.:Variationsonsequencesofarithmetic and geometric means. Zeszyty Nauk. Pol. Śl. Mat. Stosow , Wituła R., Słota D.: Intriguing it. College Math. J , 328.

8 24 P. Lorenc, R. Wituła Omówienie Autorzy prezentowanego artykułu, zainspirowani granicą1 zamieszczoną w notce[college Math. J , 328], przedstawiają tu wiele różnych nowych granic. Niektóre z nich to uogólnienia wspomnianej granicy1, a inne stanowią jedynie próbę ujęcia takich uogólnień. Obszar badań i dyskusji rozszerzono na funkcje specjalne, między innymi funkcję gamma, oraz uogólnione wielomiany Laguerre a.

Podobne dokumenty

EXAMPLES OF CABRI GEOMETRE II APPLICATION IN GEOMETRIC SCIENTIFIC RESEARCH

Anna BŁACH Centre of Geometry and Engineering Graphics Silesian University of Technology in Gliwice EXAMPLES OF CABRI GEOMETRE II APPLICATION IN GEOMETRIC SCIENTIFIC RESEARCH Introduction Computer techniques