kombinatorial matematika , a q -eksponentq -analogeksponent funktsiya , ya'ni o'ziga xos funktsiya a q - hosila. Juda ko'p .. lar bor q - lotinlar, masalan, klassik q - hosilaq -eksponentlar noyob emas. Masalan, e q ( z ) {displaystyle e_ {q} (z)} q - klassikaga mos keladigan eksponent q - hosila E q ( z ) {displaystyle {mathcal {E}} _ {q} (z)}
Ta'rif
The q -eksponent e q ( z ) {displaystyle e_ {q} (z)}
e q ( z ) = ∑ n = 0 ∞ z n [ n ] q ! = ∑ n = 0 ∞ z n ( 1 − q ) n ( q ; q ) n = ∑ n = 0 ∞ z n ( 1 − q ) n ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q ) {displaystyle e_ {q} (z) = sum _ {n = 0} ^ {infty} {frac {z ^ {n}} {[n] _ {q}!}} = sum _ {n = 0} ^ {infty} {frac {z ^ {n} (1-q) ^ {n}} {(q; q) _ {n}}} = sum _ {n = 0} ^ {infty} z ^ {n} {frac {(1-q) ^ {n}} {(1-q ^ {n}) (1-q ^ {n-1}) cdots (1-q)}}} qayerda [ n ] q ! {displaystyle [n] _ {q}!} q -faktoriy
( q ; q ) n = ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q ) {displaystyle (q; q) _ {n} = (1-q ^ {n}) (1-q ^ {n-1}) cdots (1-q)} bo'ladi q -Poxhammer belgisiq - eksponentning analogi xususiyatdan kelib chiqadi
( d d z ) q e q ( z ) = e q ( z ) {displaystyle chap ({frac {d} {dz}} ight) _ {q} e_ {q} (z) = e_ {q} (z)} Bu erda chapdagi lotin q - hosilaq -ning hosilasi monomial
( d d z ) q z n = z n − 1 1 − q n 1 − q = [ n ] q z n − 1 . {displaystyle chap ({frac {d} {dz}} ight) _ {q} z ^ {n} = z ^ {n-1} {frac {1-q ^ {n}} {1-q}} = [n] _ {q} z ^ {n-1}.} Bu yerda, [ n ] q {displaystyle [n] _ {q}} q -qavsliq -ekspensial funktsiya, qarang Ekston (1983) harvtxt xatosi: maqsad yo'q: CITEREFExton1983 (Yordam bering ) , Ismoil va Chjan (1994) harvtxt xatosi: maqsad yo'q: CITEREFIsmailZhang1994 (Yordam bering ) , Suslov (2003) harvtxt xatosi: maqsad yo'q: CITEREFSuslov2003 (Yordam bering ) va Cieslinski (2011) harvtxt xatosi: maqsad yo'q: CITEREFCieslinski2011 (Yordam bering ) .
Xususiyatlari
Haqiqatdan q > 1 {displaystyle q> 1} e q ( z ) {displaystyle e_ {q} (z)} butun funktsiya ning z {displaystyle z} q < 1 {displaystyle q <1} e q ( z ) {displaystyle e_ {q} (z)} | z | < 1 / ( 1 − q ) {displaystyle | z | <1 / (1-q)}
Teskari tomonga e'tibor bering, e q ( z ) e 1 / q ( − z ) = 1 {displaystyle ~ e_ {q} (z) ~ e_ {1 / q} (- z) = 1}
Qo'shimcha formulalar Agar x y = q y x {displaystyle xy = qyx} e q ( x ) e q ( y ) = e q ( x + y ) {displaystyle e_ {q} (x) e_ {q} (y) = e_ {q} (x + y)}
Munosabatlar
Uchun − 1 < q < 1 {displaystyle -1 E q ( z ) . {displaystyle E_ {q} (z).} asosiy gipergeometrik qatorlar ,
E q ( z ) = 1 ϕ 1 ( 0 0 ; z ) = ∑ n = 0 ∞ q ( n 2 ) ( − z ) n ( q ; q ) n = ∏ n = 0 ∞ ( 1 − q n z ) = ( z ; q ) ∞ . {displaystyle E_ {q} (z) =; _ {1} phi _ {1} chap ({scriptstyle {0 usop}}},;, zight) = sum _ {n = 0} ^ {infty} {frac { q ^ {inom {n} {2}} (- z) ^ {n}} {(q; q) _ {n}}} = prod _ {n = 0} ^ {infty} (1-q ^ { n} z) = (z; q) _ {yaroqsiz}.} Shubhasiz,
lim q → 1 E q ( z ( 1 − q ) ) = lim q → 1 ∑ n = 0 ∞ q ( n 2 ) ( 1 − q ) n ( q ; q ) n ( − z ) n = e − z . {displaystyle lim _ {q o 1} E_ {q} chap (z (1-q) ight) = lim _ {q o 1} sum _ {n = 0} ^ {infty} {frac {q ^ {inom { n} {2}} (1-q) ^ {n}} {(q; q) _ {n}}} (- z) ^ {n} = e ^ {- z}. ~} Dilogaritma bilan aloqasi e q ( x ) {displaystyle e_ {q} (x)}
e q ( x ) = ( ∏ k = 0 ∞ ( 1 − q k ( 1 − q ) x ) ) − 1 . {displaystyle e_ {q} (x) = left (prod _ {k = 0} ^ {infty} (1-q ^ {k} (1-q) x) ight) ^ {- 1}.} Boshqa tarafdan, jurnal ( 1 − x ) = − ∑ n = 1 ∞ x n n {displaystyle log (1-x) = - sum _ {n = 1} ^ {infty} {frac {x ^ {n}} {n}}} | q | < 1 {displaystyle | q | <1}
jurnal e q ( x ) = − ∑ k = 0 ∞ jurnal ( 1 − q k ( 1 − q ) x ) = ∑ k = 0 ∞ ∑ n = 1 ∞ ( q k ( 1 − q ) x ) n n = ∑ n = 1 ∞ ( ( 1 − q ) x ) n ( 1 − q n ) n = 1 1 − q ∑ n = 1 ∞ ( ( 1 − q ) x ) n [ n ] q n . {displaystyle log e_ {q} (x) = - sum _ {k = 0} ^ {infty} log (1-q ^ {k} (1-q) x) = sum _ {k = 0} ^ {infty } sum _ {n = 1} ^ {infty} {frac {(q ^ {k} (1-q) x) ^ {n}} {n}} = sum _ {n = 1} ^ {infty} { frac {((1-q) x) ^ {n}} {(1-q ^ {n}) n}} = {frac {1} {1-q}} sum _ {n = 1} ^ {infty } {frac {((1-q) x) ^ {n}} {[n] _ {q} n}}.} Chegarani olib q → 1 {displaystyle q o 1}
lim q → 1 ( 1 − q ) jurnal e q ( x / ( 1 − q ) ) = L men 2 ( x ) , {displaystyle lim _ {q o 1} (1-q) log e_ {q} (x / (1-q)) = mathrm {Li} _ {2} (x),} qayerda L men 2 ( x ) {displaystyle mathrm {Li} _ {2} (x)} dilogaritma .
Adabiyotlar
Exton , H. (1983), q-gipergeometrik funktsiyalar va ilovalar , Nyu-York: Halstead Press, Chichester: Ellis Xorvud, ISBN 0853124914 , ISBN 0470274530 , ISBN 978-0470274538 Gasper , G. & Rahmon , M. (2004), Asosiy gipergeometrik qatorlar , Kembrij universiteti matbuoti, ISBN 0521833574 Ismoil , M. E. H. (2005), Bitta o'zgaruvchida klassik va kvantli ortogonal polinomlar , Kembrij universiteti matbuoti.Ismoil , M. E. H. & Chjan , R. (1994), "Ayrim integral operatorlarning diagonalizatsiyasi", Matematikadagi yutuqlar. 108, 1-33.Ismoil , M.E.H. Rahmon , M. & Chjan , R. (1996), Ayrim integral operatorlarning diagonalizatsiyasi II, J. Komp. Qo'llash. Matematika. 68, 163-196.Jekson, F. H. (1908), "q-funktsiyalar va ma'lum farq operatori to'g'risida", Edinburg qirollik jamiyatining operatsiyalari , 46 , 253-281. 
![e_ {q} (z) = sum _ {{n = 0}} ^ {infty} {frac {z ^ {n}} {[n] _ {q}!}} = sum _ {{n = 0} } ^ {infty} {frac {z ^ {n} (1-q) ^ {n}} {(q; q) _ {n}}} = sum _ {{n = 0}} ^ {infty} z ^ {n} {frac {(1-q) ^ {n}} {(1-q ^ {n}) (1-q ^ {{n-1}}) cdots (1-q)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f224e5c169bd7d785bc27d1cb5c515c5416330d1)
![[n] _ {q}!](https://wikimedia.org/api/rest_v1/media/math/render/svg/265dcf73f45dc7d6cff172fe6b6d0f457f1c107f)
![chap ({frac {d} {dz}} ight) _ {q} z ^ {n} = z ^ {{n-1}} {frac {1-q ^ {n}} {1-q}} = [n] _ {q} z ^ {{n-1}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5454c7904c36ab46a23fd93c086675aa4f7f618)
![[n] _q](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dfd22063139d4a06ae9dfcaf79a85a2b2f751fc)
![{displaystyle log e_ {q} (x) = - sum _ {k = 0} ^ {infty} log (1-q ^ {k} (1-q) x) = sum _ {k = 0} ^ {infty } sum _ {n = 1} ^ {infty} {frac {(q ^ {k} (1-q) x) ^ {n}} {n}} = sum _ {n = 1} ^ {infty} { frac {((1-q) x) ^ {n}} {(1-q ^ {n}) n}} = {frac {1} {1-q}} sum _ {n = 1} ^ {infty } {frac {((1-q) x) ^ {n}} {[n] _ {q} n}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46d1c45c044490123c81aab2a12e18ea8c32c329)
