ZESZYTY NAUKOWE POLITECHNIKI ŚLĄSKIEJ 2013 Seria: MATEMATYKA STOSOWANA z. 3 Nr kol. 1899 Piotr LORENC, Roman WITUŁA Institute of Mathematics Silesian University of Technology SOME INTRIGUING LIMITS CONTINUATION Summary. Wituła and Słota in[college Math. J. 422011, 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. 422011, 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. 2010 Mathematics Subject Classification: 40A05. Corresponding author: R. WitułaRoman.Witula@polsl.pl. Received: 06.11.2013 r. Piotr Lorenc will be net year the MSc graduate student in Mathematics.
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 1 1+ 1... 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 1+ 1 2 ep 5 ; 4 12 e 1+ 1 e 2 ; 5 2 e e 1+ 2 1 11 e. 6 24
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 1 2 1 n n+1 n ep βln 1+ 1 β 1 1 2β + 1 3β 2 2+o 1 2 2β 1 2α + 1 1 1 3α 2 2 3β 2 2+o 2 1 1 2β +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 1+ 1 1 e 1+t t, t 0 + t n+1 n,
20 P. Lorenc, R. Wituła 2 e e 1+ 2 1 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 1+ 1 1 1 2 + 1 1 3 2+o 2 ln 1 1 2 + 1 1 2 3 2+o 2 1 5 1 12 +o, 1 5 1 ep 12 +o 5. 12 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 1+ 1 1+ 1 αe + α foreveryα R.
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 8 +1 3 1 n a n+1 n+1 n 1 n a n+1 n+1 n k n1 a 3 a 2 a 4 2 24 +a 3 +1 2 2 3+... fora1andfromthefollowingdecompositionwhichisaspecialformofbinomial series: 1+ 1 α1+ α 1 α α +o α weget 1+ 1 1+ 1 α 1+ 1 α α 1+ 1 α +e 1+ 1 α α e + e 1+ 1 + 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 +1 2 1 2 +1 whereconstantsα k satisfytherecurrencerelation k j0 m 2 + α k 2k+O 1 2m+2 k0 α j 2k 2j+1 1 22k+3 B 2k+2 22k+12k+2 fork0,1,2,...andb n denotesthen-thbernoullinumberseee.g.[9].,
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 1 2 12z 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 1 240. 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 2 1 3 12a 48 2 +241 az+4z 2 z forz C\,0],a C,whereRe nzdenotes θ n z cos 2 andθ: argz π,π].
Some intriguing its continuation 23 References 1. Assche van W.: Weighted zero distribution for polynomials orthogonal on an infinite interval. SIAM J. Math. Anal. 161985, 1317 1334. 2. Assche van W.: Erratum to Weighted zero distribution for polynomials orthogonal on an infinite interval. SIAM J. Math. Anal. 322001, 1169 1170. 3. Borwein D., Borwein J.M., Crandall R.E.: Effective Laguerre asymptotics. SIAM J. Numer. Anal. 462008, 3285 3312. 4. 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 642013, 403 416. 6. 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 1964. 9.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 1975. 13. Tricomi F., Erdelyi A.: The asymptotic epansions of a ratio of gamma functions. Pacific J. Math. 11951, 133 142. 14.WitułaR.,JamaD.,NowakI.,OlczykP.:Variationsonsequencesofarithmetic and geometric means. Zeszyty Nauk. Pol. Śl. Mat. Stosow. 12011, 81 98. 15. Wituła R., Słota D.: Intriguing it. College Math. J. 422011, 328.
24 P. Lorenc, R. Wituła Omówienie Autorzy prezentowanego artykułu, zainspirowani granicą1 zamieszczoną w notce[college Math. J. 422011, 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.