Jackson の第2 種q-Bessel 関数の 精度保証付き数値計算法

Verified computational methods for Jackson's 2nd q-Bessel function.

Transcript

1. Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷ ਫ਼౓อূ෇͖਺஋ܭࢉ๏ ۚઘେհ (ૣҴాେֶ M1)1, ؙ໺݈Ұ ୈ 1 ճ ਫ਼౓อূ෇͖਺஋ܭࢉͷ࣮໰୊΁ͷԠ༻ݚڀूձ, ๺۝भࠃࡍձٞ৔ 2017 ೥ 12 ݄ 9-10 ೔ 1https://github.com/Daisuke-Kanaizumi/q-special-functions ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 1 / 40
2. ຊൃදͷྲྀΕ 1 ݚڀഎܠ 2 Jackson ͷୈ 2 छ q-Bessel ؔ਺ 3 ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ੵ෼Λ༻͍Δํ๏ DE ެࣜʹΑΔํ๏ ઴ۙల։ʹΑΔํ๏ 4 ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͱͷൺֱ) 5 ຊݚڀͷ·ͱΊͱࠓޙͷ՝୊ ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 2 / 40
3. ݚڀഎܠ ݚڀഎܠ ͜Ε·Ͱ༷ʑͳಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ͕։ൃ͞Ε͖͕ͯͨ (Yamamoto-Matsuda (2005), Oishi (2008), Kashiwagi (kv ϥΠϒϥϦ), Yamanaka-Okayama-Oishi (2017), · · · ), Մੵ෼ܥ౳ͰݱΕΔ q-ಛघؔ਺ͷ ਫ਼౓อূ෇͖਺஋ܭࢉ๏͸·ͩͳ͍. q-ಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏Λཱ֬͢ΔͨΊ, Jackson ͷୈ 2 छ q-Bessel ؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉͨ͠. q-ಛघؔ਺: ಛघؔ਺ͷ q-ྨࣅ q ྨࣅ a aງాྑ೭, ౉ลܟҰ, ঙ࢘ढ़໌, ࡾொউٱ (2004). ܈࿦ͷਐԽ, ୅਺ֶඦՊ, I, ே૔ॻళ. ύϥϝʔλ q ΛՃ͑ΔҰൠԽ q → 1 ͱͨ͠ͱ͖ݩʹ໭Δ ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 3 / 40
4. ݚڀഎܠ q-ಛघؔ਺ q-ಛघؔ਺͸ύϥϝʔλ q ΛՃ͑ͨಛघؔ਺ͷ֦ு൛Ͱ͋Γ, q-ඍ෼΍ q-ੵ෼Λ ࢖͏ q-ղੳֶʹద߹͢ΔΑ͏ʹఆٛ͞ΕΔ (จࣈ q Λ࢖͏͚ͩͷؔ਺͸আ͘)23. q-ඍ෼ Dqf(x) := f(x) − f(qx) x(1 − q) q-ੵ෼ ∫ 1 0 f(t)dqt := (1 − q) ∞ ∑ n=0 f(qn)qn q-ղੳֶ: ۃݶΛͳΔ΂͘࢖Θͳ͍ղੳֶͷҰͭ 4 2ງాྑ೭, ౉ลܟҰ, ঙ࢘ढ़໌, ࡾொউٱ (2004). ܈࿦ͷਐԽ, ୅਺ֶඦՊ I, ே૔ॻళ. 3Ikebe, S., Graphics Library of Special Functions. http://math-functions-1.watson.jp/index.html 4Kac, V., Cheung, P. (2001). Quantum Calculus. Springer Science & Business Media. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 4 / 40
5. Jackson ͷୈ 2 छ q-Bessel ؔ਺ Jackson ͷୈ 2 छ q-Bessel ؔ਺ ࣍ͷؔ਺Λ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͱ͍͏: J(2) ν (x; q) := (qν+1; q)∞ (q; q)∞ ( x 2 )ν 0 ϕ1 ( −; qν+1, q, − qν+1x2 4 ) , x ∈ C. r ϕs ͸࣍ͷΑ͏ʹఆٛ͞ΕΔ q-௒زԿؔ਺Ͱ͋Δ (r, s ∈ Z≥0 , l = 1 + s − r): r ϕs (α1 , · · · , αr ; β1 , · · · , βs ; q, z) := ∞ ∑ n=0 r ∏ i=1 (αi ; q)n [ (−1)nq n(n−1) 2 ]l zn s ∏ j=1 (βj ; q)n (q; q)n . ҎԼͷ৐ੵͰఆٛ͞ΕΔه߸Λ q-Pochhammer ه߸ͱ͍͏ (a ∈ C, |q| < 1): (a; q)n := n−1 ∏ k=0 (1 − aqk), (a; q)0 := 1, (a; q)∞ := lim n→∞ (a; q)n . q-Painlev´ e III ܕํఔࣜͷಛघղΛهड़͢Δ (Kajiwara-Ohta-Satsuma (1995)). ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 5 / 40
6. Jackson ͷୈ 2 छ q-Bessel ؔ਺ ݚڀͷҙٛ q-ಛघؔ਺͸ q-Painlev´ e ํఔࣜͳͲ༷ʑͳํఔࣜͷղͱͯ͠ݱΕΔ 567. q-ಛघؔ਺ͷੑ࣭ (ྵ఺, ෆಈ఺, ઴ۙతڍಈͳͲ) Λݚڀ͢Δʹ͸, q-ಛघ ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏͕ॏཁʹͳΓ͏Δ. طଘެࣜͷ਺஋తݕূʹ΋࢖͑Δ. Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉΛߦ͕ͬͨ, ͦͷલʹ Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ʹ͍ͭͯݟ͍ͯ͘. 5Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y. (2004). Hypergeometric Solutions to the q-Painlev´ e Equations. International Mathematics Research Notices, 2004(47), 2497-2521. 6Kemp, A. W. (1997). On Modified q-Bessel Functions and Some Statistical Applications. Advances in Combinatorial Methods and Applications to Probability and Statistics, 451-463. Birkh¨ auser Boston. 7Կ݈ࢤ, ᝛ࡾ࿠, ๺ࠜ༃࢙. (2007). ཭ࢄ֬཰աఔͱ q-௒زԿؔ਺. ೔ຊԠ༻਺ཧֶձ࿦จࢽ, 17(4), 463-468. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 6 / 40
7. Jackson ͷୈ 2 छ q-Bessel ؔ਺ Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ ਺஋ੵ෼Λ༻͍Δํ๏ (Kashiwagi, kv ϥΠϒϥϦ”a) Jν (x) = (x/2)ν √ πΓ(ν+1/2) ∫ π 0 cos(x cos t) sin2ν tdt ઴ۙల։Λ༻͍Δํ๏ (Oishi, 2008) ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ (Yamamoto-Matsuda, 2005) aദ໦խӳ, kv - C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦ http://verifiedby.me/kv/index.html ఏҊख๏ ަ୅ڃ਺Λ༻͍Δํ๏ ઴ۙల։Λ༻͍Δํ๏ ਺஋ੵ෼Λ༻͍Δํ๏ (kv ϥΠϒϥϦ & DE ެࣜ) ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 7 / 40
8. ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ຊख๏Ͱ͸ަ୅ڃ਺ʹؔ͢ΔҎԼͷఆཧΛ༻͍Δ. ఆཧ (Leibniz) ਺ྻ {pn }∞ n=0 ͕ lim n→∞ pn = 0 Λຬͨ͢୯ௐݮগͳਖ਼਺ྻͳΒ͹, ަ୅ڃ਺ ∑ ∞ n=0 (−1)npn ͸ऩଋ͢Δ. ܥ (ަ୅ڃ਺ͷଧ੾Γޡࠩ) s := ∑ ∞ n=0 (−1)npn , sN := ∑ N n=0 (−1)npn ͱ͓͘ͱ͕࣍੒Γཱͭ: |s − sN | ≤ pN+1. ަ୅ڃ਺ͷੑ࣭Λ Jackson ͷୈ 2 छ q-Bessel ؔ਺ʹద༻Ͱ͖Δ͔ߟ͑Δ. J(2) ν (x; q) := (qν+1;q)∞ (q;q)∞ ( x 2 ) ν ∑ ∞ n=0 (−1)nqn(n−1) ( x2qν+1 4 ) n (qν+1;q)n(q;q)n . ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 8 / 40
9. ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏    0 < q < 1 x, ν ∈ R |qν| < 1, x2 < 4q ʹ੍ݶͯ͠, Jackson ͷୈ 2 छ q-Bessel ؔ਺ J(2) ν (x; q) := (qν+1;q)∞ (q;q)∞ ( x 2 )ν ∑ ∞ n=0 (−1)nqn(n−1) ( x2qν+1 4 ) n (qν+1;q)n(q;q)n ʹݱΕΔ ∑ ∞ n=0 (−1)nqn(n−1) ( x2qν+1 4 ) n (qν+1;q)n(q;q)n ʹରͯ͠લड़ͷަ୅ڃ਺ͷੑ࣭͕ద༻ Ͱ͖Δ͜ͱΛࣔ͢. ͦͷͨΊʹ, ڃ਺ͷத਎Λ dn ͱ͓͍ͨͱ͖ʹ dn ͷઈର஋͕ ୯ௐݮগ͢ΔͨΊͷ৚݅ΛٻΊΔ. ͭ·Γ, |cn | = dn dn−1 ≤ 1 ͕੒ΓཱͭͨΊͷ n ʹؔ͢Δ৚݅ΛٻΊΔ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 9 / 40
10. ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ |cn| ≤ 1 ⇐⇒ x2q2n+ν−1/4 ≤ (1 − qn)(1 − qν+n) ⇐⇒ Q2qν ( 1 − x2 4q ) − Q(1 + qν ) + 1 ≥ 0 (Q := qn). ∴ Q ≤ 1+qν − √ (1+qν )2−qν ( 4− x2 q ) qν ( 1− x2 4q ) =: A. ∴ n ≥ log A log q . ͭ·Γ log A log q Ҏ্ͷ n Ͱ͸ |cn | = | dn dn−1 | ≤ 1 ͱͳΔ. Αͬͯ {dn } ͸ͦͷઈର ஋͕୯ௐݮগ͢Δަ୅਺ྻͱͳΔ. ैͬͯަ୅ڃ਺ͷੑ࣭͕ద༻Ͱ͖Δ. ∑ ∞ n=0 (−1)nqn(n−1) ( x2qν+1 4 ) n (qν+1;q)n(q;q)n ͷଧ੾Γޡ͕ࠩධՁͰ͖ͨ. ޙ͸ (qν+1;q)∞ (q;q)∞ ͷଧ੾ΓޡࠩΛධՁͰ͖Ε͹ Jackson ͷୈ 2 छ q-Bessel ؔ਺ J(2) ν (x; q) := (qν+1; q)∞ (q; q)∞ ( x 2 ) ν ∞ ∑ n=0 (−1)nqn(n−1) ( x2qν+1 4 ) n (qν+1; q)n(q; q)n Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖Δ. ަ୅ڃ਺ͷੑ࣭͕ద༻Ͱ͖ΔΑ͏ʹ (qν+1;q)∞ (q;q)∞ ΛมܗͰ͖ͳ͍ͩΖ͏͔? ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 10 / 40
11. ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ͜͜Ͱ, ิॿతʹҎԼͷؔ਺Λ༻͍Δ. ఆٛ (q-exponential ؔ਺) eq (z) := ∑ ∞ n=0 zn (q;q)n , |z| < 1 ͜ͷؔ਺͸ҎԼͷΑ͏ͳੑ࣭Λ࣋ͭ͜ͱ͕஌ΒΕ͍ͯΔ 8. ఆཧ (Euler) eq (z) = 1 (z; q)∞ ఆཧ (Karpelvich) eq (z) = 1 (q; q)∞ ∞ ∑ n=0 (−1)nqn(n+1)/2 (q; q)n (1 − zqn) ্ͷ 2 ࣜΑΓ, |z| < 1 ͷͱ͖, ࣍ͷ౳͕ࣜ੒Γཱͭ. (z;q)∞ (q;q)∞ = 1/ ∑ ∞ n=0 (−1)nqn(n+1)/2 (q;q)n(1−zqn) . 8Olshanetsky, M. A., Rogov, V. B. (1995). The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions. arXiv preprint q-alg/9509013. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 11 / 40
12. ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ (z;q)∞ (q;q)∞ = 1/ ∑ ∞ n=0 (−1)nqn(n+1)/2 (q;q)n(1−zqn) . ∑ ∞ n=0 (−1)nqn(n+1)/2 (q;q)n(1−zqn) ʹରͯ͠લड़ͷަ୅ڃ਺ͷఆཧΛద༻Ͱ͖Δ. (ূ໌͸ઌ΄Ͳѻͬͨ ∑ ∞ n=0 (−1)nqn(n+1)/2x2n (qν+1;q)n(q;q)n ͱಉ༷Ͱ͋Δ.) (z;q)∞ (q;q)∞ = 1/ ∑ ∞ n=0 (−1)nqn(n+1)/2 (q;q)n(1−zqn) ͷଧ੾ΓޡࠩΛධՁͰ͖Δ. z = qν+1 ͱ͢Ε͹ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͰݱΕͨ (qν+1;q)∞ (q;q)∞ ͷ ଧ੾ΓޡࠩΛධՁͰ͖Δ. (qν+1;q)∞ (q;q)∞ ͱ ∑ ∞ n=0 (−1)nqn(n−1) ( x2qν+1 4 ) n (qν+1;q)n(q;q)n ͷଧ੾ΓޡࠩΛධՁͰ͖ΔͷͰ, Jackson ͷୈ 2 छ q-Bessel ؔ਺ J(2) ν (x; q) := (qν+1; q)∞ (q; q)∞ ( x 2 )ν ∞ ∑ n=0 (−1)nqn(n−1) ( x2qν+1 4 )n (qν+1; q)n (q; q)n Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖Δ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 12 / 40
13. ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ਺஋࣮ݧ (Mathematica ͱͷൺֱ) Jackson ͷୈ 2 छ q-Bessel ؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉ͢ΔϓϩάϥϜΛ C++ Ͱࣗ࡞͠, Mathematica ͷܭࢉ݁Ռͱൺֱͨ͠. ࣮ݧ؀ڥ OS: Ubuntu14.04LTS, CPU: Intel Xeon(R) CPU E3-1241 v3 @ 3.50GHz × 8 ϝϞϦ: 15.6GB, kv ϥΠϒϥϦͷόʔδϣϯ: 0.4.41, gcc version 4.8.4 ࣮ݧ݁Ռ (q = 0.1, x = 0.6, ν = 2) Mathematica ͷܭࢉ݁Ռ (ۙࣅ): 0.1009999898980716 ਫ਼౓อূͷ݁Ռ (۠ؒ): [0.10099998989807004, 0.10099998989807313] ਫ਼౓อূ෇͖਺஋ܭࢉʹΑΔ݁Ռ͕ Mathematica ͷܭࢉ݁ՌΛแؚ͍ͯ͠Δ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 13 / 40
14. ݚڀ੒Ռ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ަ୅ڃ਺ͷੑ࣭Λ༻͍Δํ๏ ަ୅ڃ਺ͷੑ࣭ͱ q-exponential ؔ਺ͷม׵ެࣜΛ༻͍ͯ Jackson ͷ ୈ 2 छ q-Bessel ؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖ͨ. ͔͜͠͠ͷख๏͸    0 < q < 1 x, ν ∈ R |qν| < 1, x2 < 4q ʹ੍ݶ͠ͳ͍ͱద༻Ͱ͖ͳ͍. x, ν ʹؔ͢Δ੍ݶͳ͠ʹਫ਼౓อূ෇͖਺஋ܭࢉͰ͖ͳ͍ͩΖ͏͔? ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 14 / 40
15. ݚڀ੒Ռ ੵ෼Λ༻͍Δํ๏ ੵ෼Λ༻͍Δํ๏ x, ν ʹؔ͢Δ੍ݶͳ͠ʹ Jackson ͷୈ 2 छ q-Bessel ؔ਺Λਫ਼౓อূ෇͖਺஋ܭ ࢉ͢ΔͨΊʹੵ෼Λ༻͍Δํ๏Λ։ൃͨ͠. ͜͜Ͱ͸ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷ࣋ͭ࣍ͷੵ෼දࣔΛ࢖͏ 9: J(2) ν (x; q) = (q2ν; q)∞ 2π(qν; q)∞ (x/2)ν × ∫ π 0 ( e2iθ, e−2iθ, −ixq(ν+1)/2 2 eiθ, −ixq(ν+1)/2 2 e−iθ; q ) ∞ (e2iθqν, e−2iθqν; q)∞ dθ, (a1 , a2 , · · · , an ; q)∞ := (a1 ; q)∞ (a2 ; q)∞ · · · (an ; q)∞ , Re ν > 0. 9Rahman, M. (1987). An Integral Representation and Some Transformation Properties of q-Bessel Functions. Journal of Mathematical Analysis and Applications, 125(1), 58-71. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 15 / 40
16. ݚڀ੒Ռ ੵ෼Λ༻͍Δํ๏ J(2) ν (x; q) = (q2ν; q)∞ 2π(qν; q)∞ (x/2)ν × ∫ π 0 ( e2iθ, e−2iθ, −ixq(ν+1)/2 2 eiθ, −ixq(ν+1)/2 2 e−iθ; q ) ∞ (e2iθqν, e−2iθqν; q)∞ dθ, (a1 , a2 , · · · , an ; q)∞ := (a1 ; q)∞ (a2 ; q)∞ · · · (an ; q)∞ , Re ν > 0. q-Pochhammer ه߸ (z; q)∞ ʹ͍ͭͯ͸ҎԼͷఆཧ 1 Λ࢖͑͹ਫ਼౓อূ෇͖਺ ஋ܭࢉͰ͖ΔͷͰ, ͋ͱ͸ແݶੵͷੵ෼ΛͲ͏͢Δ͔͕໰୊ʹͳΔ. ඃੵ෼ؔ਺ Λੵ෼͠΍͍͢ܗʹมܗ͍ͯ͘͜͠ͱΛߟ͑Δ. ఆཧ 1 a aZhang. R (2008), Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series, Advances in Mathematics 217, 1588-1613 z ∈ C, 0 < q < 1 ͱ͢Δ. ਖ਼ͷ੔਺ n ʹରͯ͠ 0 < |z|qn 1−q < 1 2 Ͱ͋Δͱ͖, ͕࣍੒Γཱͭ: (z; q)∞/(z; q)n = (zqn; q)∞ = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn/(1 − q). ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 16 / 40
17. ݚڀ੒Ռ ੵ෼Λ༻͍Δํ๏ ඃੵ෼ؔ਺ͷมܗʹ͸ҎԼΛ༻͍Δ. ఆཧ 1 (࠶ܝ)a aZhang, R. (2008). Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series. Advances in Mathematics, 217(4), 1588-1613. z ∈ C, 0 < q < 1 ͱ͢Δ. ͋Δ n ∈ N ʹରͯ͠ 0 < |z|qn 1−q < 1 2 Ͱ͋Δͱ͖Ҏ Լ͕੒Γཱͭ: (z;q)∞ (z;q)n = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn 1−q ఆཧ 2 a aZhang, R. (2008). Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series. Advances in Mathematics, 217(4), 1588-1613. ఆཧ 1 ͱಉ͡ԾఆͰҎԼ͕੒Γཱͭ: (z;q)n (z;q)∞ = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn 1−q ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 17 / 40
18. ݚڀ੒Ռ ੵ෼Λ༻͍Δํ๏ ఆཧ 1, 2 ΑΓ, ∫ π 0 ( e2iθ, e−2iθ, −ixq(ν+1)/2 2 eiθ, −ixq(ν+1)/2 2 e−iθ; q ) ∞ (e2iθqν, e−2iθqν; q)∞ dθ = [ 1 ± 2|qν+n| 1 − q ]2 [ 1 ± 2qn 1 − q ]2 [ 1 ± |xq(ν+1)/2|qn 1 − q ]2 × ∫ π 0 ( e2iθ, e−2iθ, −ixq(ν+1)/2 2 eiθ, −ixq(ν+1)/2 2 e−iθ; q ) n (e2iθqν, e−2iθqν; q)n dθ ͱมܗͰ͖Δ. ͨͩ͠ n ∈ N ͸ |qν+n| 1−q < 1 2 , qn 1−q < 1 2 , |xq(ν+1)/2|qn 2(1−q) < 1 2 (ఆཧ 1,2 ͷԾఆ) Λຬͨ͢΋ͷͰ͋Γ, [a ± b] := [a − b, a + b] ͱ͢Δ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 18 / 40
19. ݚڀ੒Ռ ੵ෼Λ༻͍Δํ๏ ੵ෼Λ༻͍Δํ๏ มܗޙͷੵ෼ʹ͸ C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦͰ͋Δ”kv ϥΠ ϒϥϦ”ʹ૊Έࠐ·Ε͍ͯΔਫ਼౓อূ෇͖਺஋ੵ෼ύοέʔδΛ࢖͏. ”kv ϥΠϒϥϦ”ʹΑΔਫ਼౓อূ෇͖਺஋ੵ෼ͷྲྀΕ a aദ໦խӳ, ϕΩڃ਺ԋࢉʹ͍ͭͯ, http://verifiedby.me/kv/psa/psa.pdf ੵ෼۠ؒΛ෼ׂ͢Δ (࣮ݧͰ͸ 10 ݸʹ෼ׂ) ⇓ ඃੵ෼ؔ਺ f ʹରͯ͠৒༨߲෇͖ Taylor ల։Λߦ͏ ⇓ ֤۠෼Ͱ f ͷ૾Λ܎਺͕۠ؒͰ͋Δଟ߲ࣜͱͯ͠ಘΔ (࣮ݧͰ͸ 10 ࣍ʹࢦఆ) ⇓ ֤۠෼ͰಘΒΕͨଟ߲ࣜΛෆఆੵ෼ͯ͠ݪ࢝ؔ਺ΛಘΔ ⇓ ֤۠෼Ͱ۠ؒ୺ͷ஋Λ୅ೖͯ͠ఆੵ෼ͷ஋Λ۠ؒͱͯ͠ಘΔ ࠓճѻ͏ੵ෼͸ඃੵ෼ؔ਺͕ෳૉؔ਺ͳͷͰ, ඃੵ෼ؔ਺Λ࣮෦ͱڏ෦ʹ෼͚ͯ ͦΕͧΕʹରͯ͠ਫ਼౓อূ෇͖਺஋ੵ෼Λߦ͏. (͜͜·Ͱ͕ kv ʹΑΔํ๏) →(ޙఔ) ೋॏࢦ਺ؔ਺ܕੵ෼ެࣜ (DE ެࣜ) Λ༻͍Δ৔߹ͱൺֱ͢Δ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 19 / 40
20. ݚڀ੒Ռ ੵ෼Λ༻͍Δํ๏ ೋॏࢦ਺ؔ਺ܕੵ෼ެࣜ (DE ެࣜ) ೋॏࢦ਺ؔ਺ܕੵ෼ެࣜ (DE ެࣜ) ͱ͸, ೋॏࢦ਺ؔ਺ܕͷม਺ม׵ (DE ม׵) ͱ୆ܗެࣜΛ૊Έ߹Θͤͨ਺஋ੵ෼๏Ͱ͋Δ 10. ͲͷΑ͏ͳ DE ม׵Λࢪ͔͢͸ ੵ෼۠ؒͱඃੵ෼ؔ਺ f ͷ࣋ͭ༏ؔ਺, ͭ·Γ |f(z)| ≤ |F (z)| (∀z ∈ Dd := {z ∈ C : | Im z| < d < π/2}) ͳΔؔ਺ F (z) ͷछྨʹԠͯ͡࢖͍෼͚͕͞Ε͍ͯΔ. DE ެࣜΛ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷੵ෼දࣔʹద༻͢Δ͜ͱΛߟ͑Δ: J(2) ν (x; q) = (q2ν ; q)∞ 2π(qν ; q)∞ (x/2)ν × ∫ π 0 ( e2iθ, e−2iθ, − ixq(ν+1)/2 2 eiθ, − ixq(ν+1)/2 2 e−iθ; q ) ∞ (e2iθqν , e−2iθqν ; q)∞ dθ, (a1, a2, · · · , an; q)∞ := (a1; q)∞(a2; q)∞ · · · (an; q)∞, Re ν > 0. 10Takahashi. H, Mori. M (1974). Double Exponential Formulas for Numerical Integration. Publications of the Research Institute for Mathematical Sciences, 9(3), 721-741. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 20 / 40
21. ݚڀ੒Ռ ੵ෼Λ༻͍Δํ๏ ψDE(t) := b−a 2 tanh ( π 2 sinh(t) ) + a+b 2 , a < b ͱ͍͏ DE ม׵ʹண໨͢Δ. ͜ͷม׵ʹ͍ͭͯ͸࣍ͷఆཧ͕஌ΒΕ͍ͯΔ. ఆཧ a aOkayama. T, Matsuo. T, Sugihara. M (2009). Error Estimates with Explicit Constants for Sinc Approximation, Sinc Quadrature and Sinc Indefinete Integration. Mathematical Engineering Technical Reports, METR2009-01, University of Tokyo. DDE (d) := {z = ψDE (w) : w ∈ Dd }, h := log(4dN) N , N ≥ e/(4d) ͱ͢Δ. f ͕ DDE (d) (d ∈ (0, π 2 )) Ͱਖ਼ଇͰ, ఆ਺ K ʹରͯ͠ |f(z)| ≤ K (∀z ∈ DDE (d)) ͕੒ΓཱͭͳΒ, ͕࣍੒Γཱͭ (ͨͩ͠ C3 = 1 cos2(π 2 sin d) cos d ): ∫ b a f(x)dx−h N ∑ k=−N f(ψDE(kh))ψ′ DE (kh) ≤K(b−a) ( eπ/2 + 2C3 1−e−πe/2 ) exp(−2πd/h). ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 21 / 40
22. ݚڀ੒Ռ DE ެࣜʹΑΔํ๏ DE ެࣜʹΑΔํ๏ J(2) ν (x; q) = (q2ν; q)∞ 2π(qν; q)∞ (x/2)ν × ∫ π 0 ( e2iθ, e−2iθ, −ixq(ν+1)/2 2 eiθ, −ixq(ν+1)/2 2 e−iθ; q ) ∞ (e2iθqν, e−2iθqν; q)∞ dθ, (a1 , a2 , · · · , an ; q)∞ := (a1 ; q)∞ (a2 ; q)∞ · · · (an ; q)∞ , Re ν > 0. ͍·, ඃੵ෼ؔ਺͸͢΂ͯऩଋ͢ΔແݶੵͳͷͰ༗քͰ͋Γ, |f(z)| ≤ K (∀z ∈ DDE (d)) Λຬͨ͢. ఆ਺ K ΛܾఆͰ͖Ε͹ DE ެࣜʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉ͕Մೳ ʹͳΔ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 22 / 40
23. ݚڀ੒Ռ DE ެࣜʹΑΔํ๏ ఆ਺ K ͷධՁ ఆ਺ K ͷධՁʹ͸ҎԼΛ༻͍Δ. ఆཧ 1 (࠶ܝ)a aZhang, R. (2008). Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series. Advances in Mathematics, 217(4), 1588-1613. z ∈ C, 0 < q < 1 ͱ͢Δ. ͋Δ n ∈ N ʹରͯ͠ 0 < |z|qn 1−q < 1 2 Ͱ͋Δͱ͖Ҏ Լ͕੒Γཱͭ: (z;q)∞ (z;q)n = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn 1−q ఆཧ 2 (࠶ܝ) a aZhang, R. (2008). Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series. Advances in Mathematics, 217(4), 1588-1613. ఆཧ 1 ͱಉ͡ԾఆͰҎԼ͕੒Γཱͭ: (z;q)n (z;q)∞ = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn 1−q ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 23 / 40
24. ݚڀ੒Ռ DE ެࣜʹΑΔํ๏ ఆཧ 1, 2 ΑΓ (ζ = x + iy, n ∈ N ͸ఆཧ 1,2 ͷԾఆΛຬͨ͢ͱ͢Δ), ( e±2iζ; q ) ∞ ≤ ( 1 + 2qne∓2y 1 − q ) ( e±2iζ; q ) n ≤ ( 1 + 2qn exp ( 2 sin(π sin(d)) cos(π sin(d))+1 ) 1 − q ) ( − exp ( 2 sin(π sin(d)) cos(π sin(d)) + 1 ) ; q ) n , 1/ ( e±2iθqν ; q ) ∞ ≤ ( 1 + |qν+n| exp ( 2 sin(π sin(d)) cos(π sin(d))+1 ) 1 − q ) / ( qν exp ( − 2 sin(π sin(d)) cos(π sin(d)) + 1 )) n ͱධՁͰ͖Δ ( ( −ixq(ν+1)/2 2 e±iθ; q ) ∞ ΋ಉ༷, ζ ∈ DDE (d) ΑΓ | Im ζ| ≤ sin(π sin(d)) cos(π sin(d))+1 ͕੒Γཱ͍ͬͯΔ͜ͱʹ஫ҙ͢Δ). ఆ਺ K ͷධՁ͕Ͱ͖ͨ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 24 / 40
25. ݚڀ੒Ռ DE ެࣜʹΑΔํ๏ ਺஋࣮ݧ Jackson ͷୈ 2 छ q-Bessel ؔ਺Λ DE ެࣜʹΑͬͯਫ਼౓อূ෇͖਺஋ܭࢉ͢Δϓ ϩάϥϜΛ C++Ͱࣗ࡞͠, ੵ෼Λ༻͍Δํ๏ (kv) ͱൺֱͨ͠. ࣮ݧ؀ڥ OS: Ubuntu14.04LTS CPU: Intel Xeon(R) CPU E3-1241 v3 @ 3.50GHz × 8 ϝϞϦ: 15.6GB, kv ϥΠϒϥϦͷόʔδϣϯ: 0.4.42 ίϯύΠϥʔ: gcc 4.8.4 ࣮ݧ݁Ռ (q = 0.1, ν = 4.5, x = 60 + 100i) ਫ਼౓อূͷ݁Ռ (DE ެࣜ):([-8584953.5198317655,-8584953.5198080316])+ ([-99374452.859596462,-99374452.859525665])i ਫ਼౓อূͷ݁Ռ (kv):([-8584953.5213287343,-8584953.5183109082])+ ([-99374452.86108996,-99374452.858030959])i DE ެࣜΛ࢖ͬͯܭࢉΛͨ͠ํ͕۠ؒ෯͕খ͘͞ͳ͍ͬͯΔ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 25 / 40
26. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ઴ۙల։ʹΑΔํ๏ ͜͜Ͱ͸ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷ࣋ͭ࣍ͷ઴ۙల։Λ༻͍Δ 11: J(2) ν (x; q) = (x/2)ν( √ q; q)∞ 2(q; q)∞ ×[f(x/2, q(ν+1/2)/2; q) + f(−x/2, q(ν+1/2)/2; q)], f(x, a; q) := (iax; √ q)∞ 3 ϕ2 ( a, −a, 0 − √ q, iax ; √ q, √ q ) . q-Pochhammer ه߸ͱ q-௒زԿؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉ͢Ε͹઴ۙల։ʹ Αͬͯਫ਼౓อূ෇͖਺஋ܭࢉग़དྷΔͱ͍͏͜ͱ͕෼͔Δ. 11Chen Y., Ismail M. E. H.; Muttalib K.A. (1994), ”Asymptotics of Basic Bessel Functions and q-Laguerre Polynomials.”, Journal of Computational and Applied Mathematics, 54: 263-272 ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 26 / 40
27. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ઴ۙల։ʹΑΔํ๏ q-Pochhammer ه߸ͷਫ਼౓อূ෇͖਺஋ܭࢉʹ͸ఆཧ 1 Λ࢖͏. ఆཧ 1 (࠶ܝ)a aZhang. R (2008), Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series, Advances in Mathematics 217, 1588-1613 z ∈ C, 0 < q < 1 ͱ͢Δ. ਖ਼ͷ੔਺ n ʹରͯ͠ 0 < |z|qn 1−q < 1 2 Ͱ͋Δͱ͖, ͕࣍੒Γཱͭ: (z; q)∞ /(z; q)n = (zqn; q)∞ = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn/(1 − q). ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 27 / 40
28. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ q-௒زԿؔ਺ (ͨͩ͠, l = 1 + s − r.) r ϕs (α1 , · · · , αr ; β1 , · · · , βs ; q, z) := ∞ ∑ n=0 ∏ r i=1 (αi ; q)n [ (−1)nq n(n−1) 2 ]l zn ∏ s j=1 (βj ; q)n (q; q)n ͷਫ਼౓อূ෇͖਺஋ܭࢉ (ଧ੾ΓޡࠩͷධՁ) ʹ͸ఆཧ 3, 4 Λ࢖༻͢Δ. ఆཧ 3 (r ≤ s ͷͱ͖) T (n) := ∏ r i=1 (αi;q)n [ (−1)nq n(n−1) 2 ] l zn ∏ s j=1 (βj ;q)n(q;q)n ͱ͓͘ͱ, r ≤ s, |βj | ≤ q−N ͷͱ͖, | ∑ ∞ n=N T (n)| ≤ ∑ ∞ n=N |T (n)| ≤ C|T (N)|, C = { 1 1−D (D < 1) ∞ (D ≥ 1) , D = r ∏ i=1 ( 1 + |βi − αi |qN |1 − βi qN | ) s+1 ∏ i=r+1 |z|qNl |1 − βi qN | . ͨͩ͠, βs+1 := q. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 28 / 40
29. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ఆཧ 4 (r = s + 1 ͷͱ͖) T (n) ͸ఆཧ 3 ͱಉ͡Ͱ, r = s + 1, |βj | ≤ q−N ͷͱ͖, | ∑ ∞ n=N T (n)| ≤ ∑ ∞ n=N |T (n)| ≤ C|T (N)|, C = { 1 1−D (D < 1) ∞ (D ≥ 1) , D = |z| s ∏ i=1 ( 1 + |βi − αi |qN |1 − βi qN | ) E, E = 1 + qN |q−αs+1 | |1−qN+1| . ఆཧ 3, 4 ͷূ໌ʹ͸࣍ͷิ୊Λ༻͍Δ. ิ୊ n, N ∈ Z≥0 , n ≥ N, 0 < q < 1, c ∈ C ͱ͢Δ. |c| ≤ q−N ͷͱ͖, qn |1 − cqn| ≤ qN |1 − cqN | ͕੒Γཱͭ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 29 / 40
30. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ิ୊ͷূ໌ n ≥ N, |c| ≤ q−N ʹ஫ҙ͢Δ. |q−n − c|2 − |q−N − c|2 =q−2n − q−2N − 2|c|q−n + 2|c|q−N =(q−n − q−N )(q−n + q−N − 2|c|) ≥2(q−n − q−N )(q−N − |c|) (∵ n ≥ N) ≥0 (∵ n ≥ N, |c| ≤ q−N ). ∴ |q−n − c| ≥ |q−N − c|. ∴ 1 |q−n−c| ≤ 1 |q−N −c| . ∴ qn |1−cqn| ≤ qN |1−cqN | . ิ୊͕ࣔ͞Εͨ. 2 ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 30 / 40
31. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ఆཧ 3, 4 ͷূ໌ r ≤ s ͷͱ͖ (ఆཧ 3), l = 1 + s − r ≥ 1 Ͱ͋Δ͜ͱʹ஫ҙͯ͠, T (n + 1) T (n) = |z|qnl 1 − qn+1 ∏ r i=1 (αi ; q)n+1 ∏ s j=1 (βj ; q)n+1 ∏ s j=1 (βj ; q)n ∏ r i=1 (αi ; q)n = |z|qnl 1 − qn+1 (1 − α1 qn) · · · (1 − αr qn) (1 − β1 qn) · · · (1 − βs qn) = r ∏ i=1 ( 1 + |βi − αi |qn |1 − βi qn| ) s+1 ∏ i=r+1 |z|qnl |1 − βi qn| (∵ βs+1 := q) ≤ r ∏ i=1 ( 1 + |βi − αi |qN |1 − βi qN | ) s+1 ∏ i=r+1 |z|qNl |1 − βi qN | (∵ n ≥ N) =:D (ิ୊͸࠷ޙͷେখൺֱͰ༻͍ͨ) ∴ ∑ ∞ n=N |T (n)| ≤(ॳ߲ |T (N)|, ެൺ D ͷ౳ൺ਺ྻͷ࿨)= |T (N)| 1−D (D < 1 ͷ࣌ʹݶΔ) 2 ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 31 / 40
32. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ఆཧ 3, 4 ͷূ໌ r = s + 1 ͷͱ͖ (ఆཧ 4), l = 1 + s − r = 0 ͳͷͰ T (n + 1) T (n) = |z| 1 − qn+1 ∏ s+1 i=1 (αi ; q)n+1 ∏ s j=1 (βj ; q)n+1 ∏ s j=1 (βj ; q)n ∏ s+1 i=1 (αi ; q)n = |z| 1 − qn+1 (1 − α1 qn) · · · (1 − αs+1 qn) (1 − β1 qn) · · · (1 − βs qn) =|z| s ∏ i=1 ( 1 + |βi − αi |qn |1 − βi qn| ) ( 1 + |q − αs+1 |qn 1 − qn+1 ) ≤|z| s ∏ i=1 ( 1 + |βi − αi |qN |1 − βi qN | ) ( 1 + |q − αs+1 |qN 1 − qN+1 ) ∵ n ≥ N =:D (ิ୊͸࠷ޙͷେখൺֱͰ༻͍ͨ) ∴ ∑ ∞ n=N |T (n)| ≤(ॳ߲ |T (N)|, ެൺ D ͷ౳ൺ਺ྻͷ࿨)= |T (N)| 1−D (D < 1 ͷ࣌ʹݶΔ) 2 ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 32 / 40
33. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ઴ۙల։Λ༻͍ͳ͘ͱ΋ఆཧ 1, 3, 5 Λ૊Έ߹Θͤͯ Jackson ͷୈ 2 छ q-Bessel ؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉ͢Δ͜ͱ΋Ͱ͖Δ. ఆཧ 5 (Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷผදݱ)a aKoelink, H. T. (1993). Hansen-Lommel Orthogonality Relations for Jackson’s q-Bessel Functions. Journal of Mathematical Analysis and Applications, 175(2), 425-437. (w; q)∞ 0 ϕ1 (−; w; q, wz) = 1 ϕ1 (z; 0; q, w) ΑΓ, Jackson ͷୈ 2 छ q-Bessel ؔ਺͸࣍ͷผදݱΛ࣋ͭ: J(2) ν (x; q) = (x/2)ν (q;q)∞ 1 ϕ1 (−x2/4; 0; q, qν+1). ڃ਺ͷத਎ʹ xn Λ࣋ͨͳ͍Α͏ͳผදݱͰ͋Δ. ఆཧ 5 Λ࢖͏ํ๏ͱ઴ۙల։Λ࢖͏ํ๏͸ͲͪΒ΋ q-Pochhammer ه߸ͱ q-௒ زԿؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉ͢Δ఺͸ಉ͡Ͱ͋Δ. → ਺஋࣮ݧΛ௨ͯ͠ҧ͍ΛΈ͍ͯ͘. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 33 / 40
34. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ਺஋࣮ݧ (઴ۙల։ͱఆཧ 5 ͱͷൺֱ) Jackson ͷୈ 2 छ q-Bessel ؔ਺Λ઴ۙల։ʹΑͬͯਫ਼౓อূ෇͖਺஋ܭࢉ͢Δ ϓϩάϥϜΛ C++Ͱࣗ࡞͠, ఆཧ 5 Λ࢖ͬͯܭࢉ͢Δํ๏ͱൺֱͨ͠. C++ʹ ΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦͰ͋Δ”kv ϥΠϒϥϦ”Λ࢖༻͍ͯ͠Δ. ࣮ݧ؀ڥ OS: Ubuntu14.04LTS CPU: Intel Xeon(R) CPU E3-1241 v3 @ 3.50GHz × 8 ϝϞϦ: 15.6GB, kv ϥΠϒϥϦͷόʔδϣϯ: 0.4.42 ίϯύΠϥʔ: gcc 4.8.4 ࣮ݧ݁Ռ (q = 0.1, ν = 1.5, x = 80000 + 90000i) ઴ۙల։ͷ݁Ռ:([-4.56445403584959e+22,-4.56445403583238e+22])+ ([3.26544888256245e+23,3.26544888256455e+23])i ਫ਼౓อূͷ݁Ռ (ఆཧ 5):([-4.56445403584284e+22,-4.56445403583836e+22])+ ([3.26544888256273e+23,3.26544888256352e+23])i |x| → ∞ ͱ͚ͨͩ͠Ͱ͸͋·Γҧ͍͕ແ͍Α͏ͩ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 34 / 40
35. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ਺஋࣮ݧ (઴ۙల։ͱఆཧ 5 ͱͷൺֱ) ࠓ౓͸ ν < 0 ͱͯ͠ΈΔ. ࣮ݧ݁Ռ (q = 0.1, ν = −1.5, x = 80000 + 90000i) ઴ۙల։ͷ݁Ռ:([-2.68357252450128e+23,-2.68357252449777e+23])+ ([-2.75213451400802e+22,-2.75213451397966e+22])i ਫ਼౓อূͷ݁Ռ (ఆཧ 5):θϩআࢉൃੜ ఆཧ 5 Λ࢖ͬͨ࣌ʹ͸θϩআࢉ͕ൃੜͯ͠͠·ͬͨ. ͜Ε͸ڃ਺ͷத਎ʹ͋Δ (qν)n ͕Өڹ͍ͯ͠Δ. ఆཧ 5 (࠶ܝ, Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷผදݱ)a aKoelink, H. T. (1993). Hansen-Lommel Orthogonality Relations for Jackson’s q-Bessel Functions. Journal of Mathematical Analysis and Applications, 175(2), 425-437. J(2) ν (x; q) = (x/2)ν (q;q)∞ 1 ϕ1 (−x2/4; 0; q, qν+1). ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 35 / 40
36. ݚڀ੒Ռ ઴ۙల։ʹΑΔํ๏ ਺஋࣮ݧ (઴ۙల։) ν → −∞ ͷͱ͖, ઴ۙల։Λ༻͍Δํ๏Ͱ΋ܭࢉ͕͏·͘ߦ͔ͳ͍͜ͱ͕͋Δ. ࣮ݧ݁Ռ (q = 0.1, ν = −20.5, x = 80000 + 90000i) ઴ۙల։ͷ݁Ռ:([-inf,inf])+([-inf,inf])i ಘͨ݁Ռͷ۠ؒ෯͕ແݶେʹͳͬͯ͠·ͬͨ. J(2) ν (x; q) = (x/2)ν( √ q; q)∞ 2(q; q)∞ ×[f(x/2, q(ν+1/2)/2; q) + f(−x/2, q(ν+1/2)/2; q)], f(x, a; q) := (iax; √ q)∞ 3 ϕ2 ( a, −a, 0 − √ q, iax ; √ q, √ q ) . q-௒زԿؔ਺಺ͷ q-Pochhammer ه߸ʹ͋Δ qν ͕Өڹ͍ͯ͠Δ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 36 / 40
37. ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͱͷൺֱ) ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͱͷൺֱ) Jackson ͷୈ 2 छ q-Bessel ؔ਺Λ DE ެࣜͱ઴ۙల։ʹΑͬͯਫ਼౓อূ෇͖਺஋ ܭࢉ͢ΔϓϩάϥϜΛ C++Ͱࣗ࡞͠, Mathematica 11 Λ࢖ͬͯܭࢉ͢Δํ๏ͱ ൺֱͨ͠. Mathematica11 Ͱ͸ؔ਺ QHypergeometricPFQ Λ࢖༻ͨ͠. ࣮ݧ؀ڥ OS: Ubuntu14.04LTS CPU: Intel Xeon(R) CPU E3-1241 v3 @ 3.50GHz × 8 ϝϞϦ: 15.6GB, kv ϥΠϒϥϦͷόʔδϣϯ: 0.4.42 ίϯύΠϥʔ: gcc 4.8.4 ࣮ݧ݁Ռ (q = 0.1, ν = 1.4, x = 6000 + 1000i) ਫ਼౓อূͷ݁Ռ (DE ެࣜ):([-811903610341.59888,-811903610338.72387])+ ([-3282263156357.3467,-3282263156354.1054])i ਫ਼౓อূͷ݁Ռ (઴ۙల։):([-811903610340.45716,-811903610339.47705])+ ([-3282263156356.6651,-3282263156355.2021])i Mathematica ͷ݁Ռ (ۙࣅ): -8.119036103401538×1011+ -3.2822631563556987×1012 DE ެࣜͱ઴ۙల։ʹΑΔ݁Ռ͸ Mathematica ʹΑΔܭࢉ݁ՌΛแؚ͍ͯ͠Δ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 37 / 40
38. ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͱͷൺֱ) ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͷൺֱ) x ͷઈର஋͕খ͍͞৔߹Ͱ΋࣮ݧΛߦͬͨ. ࣮ݧ݁Ռ (q = 2−53, x = 2−53, ν = 2) ਫ਼౓อূ෇͖਺஋ܭࢉͷ݁Ռ (઴ۙల։): [3.081487911019242×10−33,3.0814879110204197×10−33]+ i[−2.6664348032491181 × 10−82, 2.6664348032491181 × 10−82] ਫ਼౓อূ෇͖਺஋ܭࢉͷ݁Ռ (DE):[3.0814879110186132×10−33, 3.0814879110204197×10−33]+i[-9.2380320625533862e×10−47, ,9.2444715835654248×10−47] Mathematica ͷ݁Ռ (ۙࣅ):3.081487911019578×10−33 ͜ͷ࣌΋ਫ਼౓อূ෇͖਺஋ܭࢉͷ݁Ռ͕ Mathematica ͷ݁ՌΛแؚ͍ͯ͠Δ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 38 / 40
39. ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͱͷൺֱ) ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͷൺֱ) Mathematica11 Λ࢖ͬͯ Jackson ͷୈ 2 छ q-Bessel ؔ਺Λܭࢉ͢Δࡍ͸ QHypergeometricPFQ ͷ୅ΘΓʹ Sum Λ࢖͏͜ͱ΋Ͱ͖Δ. q = 2ˆ(-53); nu = 2 ; x = 2ˆ(-53); N[Sum[(ʖ x*x*qˆ(nu+1)/4)ˆn*qˆ(n*(n-1))/(QPochhammer[qˆ(nu+1),q,n] *QPochhammer[q,q,n]),{n,0,Infinity}]*(x/2)ˆnu* QPochhammer[qˆ(nu+1),q,Infinity]/QPochhammer[q,q,Infinity]] ࣮ݧ݁Ռ (q = 2−53, x = 2−53, ν = 2) ਫ਼౓อূ෇͖਺஋ܭࢉͷ݁Ռ (઴ۙల։): [3.081487911019242×10−33,3.0814879110204197×10−33]+ i[−2.6664348032491181 × 10−82, 2.6664348032491181 × 10−82] ਫ਼౓อূ෇͖਺஋ܭࢉͷ݁Ռ (DE):[3.0814879110186132×10−33, 3.0814879110204197×10−33]+i[-9.2380320625533862e×10−47, ,9.2444715835654248×10−47] Mathematica ͷ݁Ռ (ۙࣅ, Sum Λ࢖༻): 3.081487911019578×10−33−1.67385337691597×10−561i ࣮਺஋ܭࢉͳͷͰڏ෦͸ 0 ͷ͸͕ͣͩ Mathematica ͩͱڏ෦ ̸= 0 ʹͳΔ. ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 39 / 40
40. ຊݚڀͷ·ͱΊͱࠓޙͷ՝୊ ຊݚڀͷ·ͱΊͱࠓޙͷ՝୊ ֤ఏҊख๏ͷੑೳ ަ୅ڃ਺ ઴ۙల։ ੵ෼ (kv & DE) |x| → ∞ NG ⊚ ◦ ν → −∞ NG ◦ NG ⊚: े෼࢖͑Δ ◦: ͋Δఔ౓࢖͑Δ ࠓޙͷ՝୊ ઴ۙల։Λ༻͍ͯ΋ ν → −∞ ͷͱ͖͸͏·͍͔͘ͳ͔ͬͨ → վྑͰ͖Δ͔? ۚઘେհʢૣҴాେֶ M1) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ 2017 ೥ 12 ݄ 9-10 ೔ 40 / 40