Flomt 7:8 3), 473 483 DOI.98/FIL38473S Publshed by Fculty of Scences nd Mthemtcs, Unvesty of Nš, Seb Avlble t: http://www.pmf.n.c.s/flomt Genelzed q-ntegls v neutces: Applcton to the q-bet functon Ahmed Slem Deptment of Bsc Scence, Fculty of Infomton Systems & Compute Scence, Octobe 6 Unvesty, S of Octobe Cty, Gz, Egypt Abstct. Let f ) be contnuous functon defned on the ntevl [, ]. In ths wo, we pply the neut lmt to genelze the q-ntegl α ln f )d q, N fo ll vlues of α R. We use ou esults to etend the defnton of the q-bet functon B q, b) nd ts devtves fo ll vlues of b nd,,,. Some esults fo the q-gmm functon e deved.. Intoducton Neutces e ddtve goups of neglgble functons tht do not contn ny constnts ecept zeo. The clculus ws developed by vn de Coput [] nd Hdmd n connecton wth symptotc sees nd dvegent ntegls. Recently, the concepts of neut nd neut lmt hve been used wdely n mny pplctons n mthemtcs, physcs nd sttstcs. The technque of neglectng ppoptely defned nfnte qunttes ws devsed by Hdmd nd the esultng fnte vlue etcted fom the dvegent ntegl s usully efeed to s the Hdmd fnte pt. Fshe gve genel pncples fo the dscdng of unwnted nfnte qunttes fom symptotc epnsons nd ths hs been eploted n the contet of dstbutons, ptcully n connecton wth the composton of dstbutons, the poduct nd the convoluton poduct of dstbutons; see [ 4]. Fshe et l. [5 8] used neutces to defne some of the specl functon see lso [9 ]). Ng nd vn Dm [, 3] ppled the neut clculus, n conjuncton wth the Hdmd ntegl, developed by vn de Coput, to quntum feld theoes, n ptcul, to obtn fnte esults fo the coeffcents n the petubton sees. They lso ppled neut clculus to quntum feld theoy, obtnng fnte enomlzton n the loop clcultons. In the second hlf of the twenteth centuy thee ws sgnfcnt ncese of ctvty n the e of the q-clculus due to pplctons of the q-clculus n mthemtcs, sttstcs nd physcs. Thee e mny of q-specl functons hve q-ntegl epesenttons q-gmm, q-bet, ncomplete q-gmm, q-bessel,...etc). The q-ntegl epesenttons hve bounded domn of convegence nd they e dvegent outsde ths domn. Bette yet, we should loo fo mthemtcl tool tht cn hndle the domn of convegence. Slem [4, 5] hs suggested tht such tool s ledy vlble n the neut clculus developed by vn Mthemtcs Subject Clssfcton. Pmy 33D5; Secondy 46F Keywods. q-ntegls; neutces; neut lmt; q-gmm functon; q-bet functon Receved: 3 Jnuy 3; Revsed: June 3; Accepted: 5 June 3 Communcted by Ljubš D.R. Kočnc Eml ddess: hmedslem74@hotml.com Ahmed Slem)
A. Slem / Flomt 7:8 3), 473 483 474 de Coput []. He ppled the neut lmt to etend the defntons of the q-gmm nd the ncomplete q-gmm functons nd the devtves to negtve ntege vlues. Contnuton on the sme mnne, Ege nd Yýldýým [6] used the neut nd the neut lmt to obtn some equltes of the q-gmm functon fo ll el vlues of. In ths wo we pply the neut lmt to genelze q-ntegls α ln f )d q N.) fo ll vlues of α, whee f ) s contnuous functon defned on the ntevl [, ].. Pelmnes nd nottons Fo the convenence of the ede, we povde n ths secton summy of the mthemtcl nottons nd defntons used n ths ppe. Thoughout of ths wo, q s postve numbe, < q < nd the defntons of q-clculus wll be ten fom the well nown boos n ths feld [7, 8]. Defnton.. Notton n q-clculus): Fo ny comple numbe, we defne [] q q q, q ; [n] q! [n] q [n ] q [] q [] q, n N.) wth [] q! nd the q-shfted fctols e defned s n ; q), ; q) n q ), n N..) The lmt, lm n ; q) n, s denoted by ; q). The eponentl functon e hs mny dffeent q-etensons, one of them s defned s E q ) q n ) n q); q)..3) Let f be functon defned on subset of el o comple plne. The q-dffeence opeto s defned by the fomul f ) f q) D q f )),, nd D q f )) f ).4) q) povded f s dffeentble t ogn. The bove opeto s now sometmes efeed to s Eule-Jcson, Jcson q-dffeence opeto o smply the q-devtve. The q-ntegton of Jcson s defned fo functon f defned on genec ntevl [, b] to be whee f )d q f )d q f )d q q) povded the sum conveges bsolutely. The q-ntegtng by pts s gven fo sutble functons f nd by f )d q,.5) f q n )q n,.6) f )D q )d q f b) b) f ) ) q)d q f )d q..7) We cll the functon f s q-ntegble on the genec ntevl [, ] f the sees q n f q n ) conveges bsolutely.
A. Slem / Flomt 7:8 3), 473 483 475 Defnton.. The q-tylo sees): Let f ) be contnuous functon on some ntevl [, b] nd c, b). Jcson ntoduced the followng q-countept of Tylo sees f ) whee [ c] n n cq ) nd [ c]. [ c] n D n q f c),, b).8) Rjovc et l. [9] poved tht f f ) s contnuous functon on [, b] nd c, b), then thee ests ˆq, ) such tht fo ll q ˆq, ), ξ, b) cn be found between c nd, whch stsfes f ) n [ c] [] q! Moeove, f c, we hve fo ll q, ) thee ests ξ, ) such tht f ) n D q f c) + [ c] n D q f ξ).9) [] q! D q f ) + n D q f ξ).) Defnton.3. Neut): A neut N s defned s commuttve ddtve goup of functons f ξ) defned on domn N wth vlues n n ddtve goup N, whee futhe f fo some f n N, f ξ) γ fo ll ξ N, then γ. The functons n N e clled neglgble functons. Defnton.4. Neut lmt): Let N be set contned n topologcl spce wth lmt pont whch does not belong to N. If f ξ) s functon defned on N wth vlues n N nd t s possble to fnd constnt c such tht f ξ) c N, then c s clled the neut lmt of f s ξ tends to nd we wte N lm ξ f ξ) c. In ths ppe, we let N be the neut hvng domn N { : < < } nd nge N the el numbes, wth the neglgble functons beng fnte lne sums of the functons λ ln, ln λ <, N). nd ll functons o) whch convege to zeo n the noml sense s tends to zeo []. 3. Genelzed q-ntegls Begn ths secton by the followng lemms, whch help to pove ou esults. Lemm 3.. [4]) Fo ll, b nd ll vlues of α, we hve α ln d q fo N nd when, we hve q )ln + b ln + ) + ) ln q b α ln b α ln [α] q + qα ln q q α ln q + ) ) + ln q q )ln b ln ), α α ln q d q b α α, α [α] q ln ln q d q, α α ln d q, α. 3.) 3.)
Lemm 3.. The neut lmt s tends to zeo of the q-ntegl ests fo ll vlues of α nd N nd Futhemoe, f q, we get A. Slem / Flomt 7:8 3), 473 483 476 N lm α ln d q α ln d q 3.3) q α ln q q α )[α] q N lm α ln d ) α + Poof. Fom Lemm 3., fo ll vlues of α nd N, we hve q α ) ) q α ) 3.4) ) ) )! 3.5) α + N lm α ln d q qα ln q ) q α ln qn lm α ln d q Usng the mthemtcl nducton would yeld N lm α ln d q q α ln q q α )[α] q + qα q α ) + q α q α ) q α ) ) ) ) ) + + q α q α ln q ) q α )[α] q + qα q α ) q α ) 3 ) 3 + q α ) 3 q α ) ) + + q α ) q α ln q q α ) ) q α )[α] q q α ) By tng the lmt of equton 3.4) s q would gve esly the equton 3.5). Lemm 3.3. The nducton yelds ) ) N lm ln d q, N. 3.6) Theoem 3.4. Let f ) be contnuous functon on the closed ntevl [, ], then the functon α ln f ) s q-ntegble on [, ] fo α > nd N. Futhemoe, fo α > m, m N, we hve the functon s q-ntegble on [, ]. α ln f ) D n q f ) n, N 3.7)
A. Slem / Flomt 7:8 3), 473 483 477 Poof. Snce f ) s contnuous functon on the segment [, ], the eteme vlue theoem sttes, t ttns ts mmum M,.e. thee ests c [, ] such tht f ) f c) M fo ll [, ]. It follows tht α ln f )d q t M α ln d q t q) α M q αn n ln q + ) conveges bsolutely fo α >, N. Snce f ) s contnuous functon on the segment [, ], then f ) hs emnde tem n q-tylo fomul.). Ths would yeld α ln f ) D n q f ) n Dm q f ξ) [m] q! q) α+m Dm q f ξ) [m] q! q nα+m) n ln q + ) conveges bsolutely fo α > m, m N nd N. α+m ln d q, < ξ < Theoem 3.5. Let f ) be contnuous functon defned on [, ]. Then the neut lmt s tends to zeo of the q-ntegl α f )d q 3.8) ests fo α > m, m,,,..., α,,,..., m + nd Poof. We hve N lm α f )d q α f )d q α f ) α f ) D n q f ) D n q f ) n d q + n d q + D n q f ) D n q f ) α [α + n] q. 3.9) D n α q f ) D f ) n n q f ) α d q + [α + n] q D n q f ) α+n [α + n] q, α,,,, m + α+n d q Snce f ) s contnuous functon on the segment [, ], Theoem 3.4 tells tht the fst q-ntegl on the ght hnd sde conveges bsolutely s nd the lst sum conssts of tems tend to zeo nd the est s lne sum of λ, λ < ) neglgble functon nd hence the poof s complete. Theoem 3.6. Let f ) be contnuous functon defned on [, ]. Then the neut lmt s tends to zeo of the q-ntegl α ln f )d q, N 3.) ests fo α > m, m,,,..., α,,,..., m +
Poof. The q-ntegl 3.) cn be spltted s α ln f )d q + A. Slem / Flomt 7:8 3), 473 483 478 D n q f ) α ln f ) D n q f ) n d q D n α+n ln q f ) d q + α+n ln d q It s obvous tht the second q-ntegl conveges bsolutely fo ll vlues of α, nd fo α > m, m N nd N s, the fst q-ntegl conveges bsolutely by usng Theoem 3.4. Also the Lemm 3. tells tht the neut lmt of the lst q-ntegl ests s. Ths completes the poof. Theoem 3.7. Let f ) be contnuous functon defned on the ntevl [, ]. Then the functon s q-ntegble on [, ] fo m N nd N. m m ln f ) D n q f ) n 3.) Poof. Snce f ) s contnuous functon on the segment [, ], t stsfes.). It follows tht m m ln f ) D n q f ) n [n] q! Dm+ q f ξ) ln d q, < ξ < [m + ] q! q) Dm+ q f ξ) q n n ln q + ) [m + ] q! conveges bsolutely fo m N nd N. Theoem 3.8. Let f ) be contnuous functon defned on the ntevl [, ]. Then ests fo N. Poof. The q-ntegl 3.) cn be spltted s ln f )d q N lm ln f )d q 3.) + f ) ln [ f ) f )]d q ln d q + f ) ln d q The lmt of the fst q-ntegl on the ght hnd sde s s beng convegent by the pevous Theoem. Lemm 3.3 poves tht the neut lmt of the lst q-ntegl ests fo N. Theoem 3.9. Let f ) be contnuous functon defned on the ntevl [, ]. Then ests fo m N nd N. N lm m ln f )d q 3.3)
Poof. The q-ntegl 3.3) cn be spltted s A. Slem / Flomt 7:8 3), 473 483 479 m ln f )d q + D n q f ) D n q f ) m + m m ln f ) D n q f ) n d q n m ln d q + Dm q f ) [m] q! n m ln d q + Dm q f ) [m] q! ln d q ln d q The lmt of the fst q-ntegl on the ght hnd sde s s beng convegent by Theoem 3.7 fo ll vlues of m N nd N. The second nd thd ntegls convege bsolutely fo ll m,, usng Lemm 3.. Futhe, fom Lemms 3. nd 3.3, we see tht the neut lmt of the q-ntegl n the lst two tems on the ght hnd sde est. Ths ends the poof. The pevous theoems cn be outlned s follow: Theoem 3.. Let f ) be contnuous functon defned on the ntevl [, ]. Then ests fo ll el vlues of α nd N. N lm α ln f )d q 3.4) 4. On the q-gmm functon nd ts devtves The q-gmm functon s defned by q-ntegl epesentton [7, 8] Γ q ) Moeove, t hs the ecuence elton t E q qt)d q t, >. 4.) Γ q + ) [] q Γ). 4.) The q-gmm functon cn be defned fo < nd,,, by usng the fome ecuence fomul. In ptcul, t follows tht f > nd, then nd by q-ntegtng by pts we hve Γ q ) [] q Γ q + ) [] q t D q Eq t) ) d q t 4.3) Γ q ) t E q qt) ) d q t + q) q. 4.4) Moe genelly, t s esy to pove by mthemtcl nducton tht f > n, n N nd,,,, n+, the q-gmm functon hs the fom Γ q ) n t ) q +) E q qt) [] q! t n d qt + q) ) q +) q; q) q + ). 4.5)
A. Slem / Flomt 7:8 3), 473 483 48 It hs been shown n [4] tht the q-gmm functon 4.) s defned by the neut lmt Γ q ) N lm fo,,,..., nd ths functon s lso defned by neut lmt Γ q m) N lm t m E q qt)d q t t E q qt)d q t, 4.6) t m E q qt) m ) q +) [] q! t d q t + q) m+ ) q +) mm+) q; q) q m ) + )m q q) m+ ln q). 4.7) q; q) m ln q fo m N nd when m t hs the fom Γ q ) N lm t E q qt)d q t t [ E q qt) ] d q t + q) ln q). 4.8) ln q It ws poved lso n [4] the estence of th devtve of the q-gmm functon nd defned t fo ll vlues of by the equton Γ ) q ) N lm t ln te q qt)d q t,,. 4.9) The utho n [4] gve the fom of the th devtves of the q-gmm functon t specl cses. We use ou esults n Lemms 3. nd 3.3 to gve genel fom fo th devtves of the q-gmm functon s follows. Fo > n, n N nd N Γ ) q ) N lm Fo nd N Fo n, n N nd N + ln q n ) q +) ++ q [ + ] q[] q! t ln te q qt)d q t n t ln te q qt)d q t + t ln t E ) q +) q qt)d q t t [] q! q + ) ) ) j j. 4.) q + j Γ ) q ) N lm Γ ) q n) N lm + ln q n ) q +) + n q [ n] q[] q! t ln te q qt)d q t t ln te q qt)d q t + j t ln t [ E q qt) ] d q t. 4.) t n ln te q qt)d q t n t n ln te q qt)d q t + t n ln t E ) q +) q qt)d q t t [] q! q n ) ) ) j j. 4.) q n j j
5. The q-bet functon nd ts devtves A. Slem / Flomt 7:8 3), 473 483 48 The q-bet functon s defned by q-ntegl epesentton [7, 8] B q, b) The q-bnoml theoem would yeld the functon f ) q; q) q ; q) b q; q) q ; q) d q, b >,,,,... 5.) q ; q) n q n q; q) n n, < q 5.) s contnuous functon on the ntevl [,] so we cn pply ou esults n secton thee to defne the q-bet functon nd ts devtves fo ll vlues of b s follow: Defnton 5.. The q-nlogue of the bet functon s defned fo ll vlues of b nd,,, by B q, b) N lm b q; q) q d q. 5.3) ; q) If we wnt to me the detls of ths defnton moe pecse, by usng Lemms 3. nd 3.3, we edefne s follow: Fo b > m, m N; b,,,, m + nd N B q, b) N lm Fo b nd,,, b q; q) q ; q) d q b q; q) q ; q) n q n q ; q) q; q) n n d q ; q) n q n q +. 5.4) [n + b] q q; q) n B q, ) N lm q; q) ) q; q d q q) ; q) q d q. 5.5) ; q) Fo b m, m N nd,,, B q, m) N lm m q; q) q ; q) m q; q) q d q ; q) m q ; q) n q n n q; q) n d q + q ; q) n q n [n m] q q; q) n. 5.6) Defnton 5.. The th ptl devtve wth espect to b of the q-bet functon s defned fo ll vlues of b nd,,, s Bq b, b) ) N lm b ln q; q) q d q, N. 5.7) ; q) Also, the detls of ths defnton cn be ognzed s follow:
A. Slem / Flomt 7:8 3), 473 483 48 Fo b > m, m N; b,,,, m + ;,,, nd N Bq b, b) ) N lm b ln q; q) q d q ; q) b ln q; q) q ; q) n q n q n ; q) q; q) n d q + ln q q ; q) n q +)n+b q n+b ) ) ). 5.8) q [n + b] qq; q) n q n+b Fo b ;,,, nd N Bq b, b) ) b N lm ln q; q) ) q; q d q ln q) ; q) q d q. 5.9) ; q) Fo b m, m N;,,, nd N Bq b, b) ) b m N lm m ln q; q) q ; q) + ln q q ; q) n q +)n m q [n m] qq; q) n m ln q; q) q d q ; q) m q ; q) n q n n q; q) n d q q n m ) ) q n m ). 5.) Rem 5.3. It s nown tht the neut lmt, f t ests, s unque nd t s pecsely the sme s the odny lmt, f t ests. We cn vefy ths by usng the equton 5.8) s follows B q, b) b q; q) q ; q) q ; q) n q n [n + b] q q; q) n [b] q ϕ q, q b ; q b+ ; q, q ) q ; q) n q n n q; q) n d q ; q) n q n q + [n + b] q q; q) n q ; q) n q b ; q) n q n q; q) n q b+ ; q) n whee ϕ s the bsc Guss hypegeometc functon whch t hs the well nown dentty ϕ, b; c; q, b c) c; q) b c; q) c; q) b c; q) c b <. Usng ths dentty would yeld the well nown defnton of the q-bet functon [6] B q, b) q)q; q) q +b ; q) Γ q)γ q b),, b,,,. 5.) q ; q) q b ; q) Γ q + b) Refeences [] J.G. vn de Coput, Intoducton to the neut clculus, J. Anl. Mth. 7 959-96) 9 398. [] B. Fshe, K. Ts, The convoluton of functons nd dstbutons, J. Mth. Anl. Appl. 36 5) 364 374. [3] B. Fshe, A non-commuttve neut poduct of dstbutons, Mth. Nch. 8 98) 7 7.
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