乗積公式と積分によるq-gamma 関数の 精度保証付き数値計算

Verified computational methods for the q-gamma function.

Transcript

1. ৐ੵެࣜͱੵ෼ʹΑΔq-gamma ؔ਺ͷ ਫ਼౓อূ෇͖਺஋ܭࢉ ۚઘେհ (ૣҴాେֶ, M1)1, ؙ໺݈Ұ (ૣҴాେֶ) 2017 ೥౓Ԡ༻਺ֶ߹ಉݚڀूձ ཾ୩େֶ 2017 ೥ 12 ݄ 14-16 ೔ 1https://github.com/Daisuke-Kanaizumi/q-special-functions ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 1 / 33
2. ຊൃදͷྲྀΕ 1 ݚڀഎܠ 2 q-gamma ؔ਺ 3 ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ ੵ෼ʹΑΔํ๏ 4 ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͱͷൺֱ) 5 ຊݚڀͷ·ͱΊͱࠓޙͷ՝୊ ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 2 / 33
3. ݚڀഎܠ ݚڀഎܠ ͜Ε·Ͱ༷ʑͳಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ͕ݚڀ͞Ε͖͍ͯͯΔ͕ (Yamamoto-Matsuda (2005), Oishi (2008), Kashiwagi (kv ϥΠϒϥϦ), Yamanaka-Okayama-Oishi (2017), · · · ), Մੵ෼ܥ౳ͰݱΕΔ q-ಛघؔ਺ (ಛघؔ਺ͷ q ྨࣅ) ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏͸·ͩͳ͍. q-ಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏Λཱ֬͢ΔͨΊ, q-gamma ؔ਺ͷਫ਼౓ อূ෇͖਺஋ܭࢉΛߦͳͬͨ. ਫ਼౓อূ෇͖਺஋ܭࢉ ۙࣅ஋ͷܭࢉΛ͢Δͱಉ࣌ʹܭࢉ݁Ռͷ (਺ֶతʹ) ݫີͳޡࠩධՁ΋ߦ͏਺஋ ܭࢉ๏ͷ͜ͱ. ਅ஋ΛؚΉ۠ؒΛ݁Ռͱͯ͠ग़ྗ͢Δ. q ྨࣅ ύϥϝʔλ q ΛՃ͑ΔҰൠԽ q → 1 ͱͨ͠ͱ͖ݩʹ໭Δ ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 3 / 33
4. ݚڀഎܠ ۠ؒԋࢉ ਫ਼౓อূ෇͖਺஋ܭࢉͰ͸” ۠ؒԋࢉ” ͱ͍͏ٕज़ʹΑͬͯ਺஋ܭࢉͷࡍʹੜ͡ ΔޡࠩΛ೺Ѳ͍ͯ͠Δ. ۠ؒԋࢉ a aେੴਐҰ. (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. ۠ؒԋࢉΛߦ͏ࡍ͸਺Λด۠ؒʹஔ͖׵͑ͯԼهͷϧʔϧʹै͍ܭࢉ͍ͯ͠Δ. (¯ x ্͕ݶ, x ͕Լݶ,[x] = [x, ¯ x] ͱ͢Δ) Ճࢉ: [x] + [y] = [x + y, ¯ x + ¯ y] ݮࢉ: [x] − [y] = [x − ¯ y, ¯ x − y] ৐ࢉ: [x] × [y] = [min(xy, x¯ y, y¯ x, ¯ xy), max(xy, x¯ y, y¯ x, ¯ xy)] আࢉ: [x]/[y] = [min(x/y, x/¯ y, ¯ x/y, ¯ x/¯ y), max(x/y, x/¯ y, y/¯ x, ¯ x/¯ y)] (ͨͩ͠ [y] ͸ 0 Λؚ·ͳ͍۠ؒͱ͢Δ) ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 4 / 33
5. ݚڀഎܠ ਫ਼౓อূ෇͖਺஋ܭࢉͰѻ͏ޡࠩ ਫ਼౓อূ෇͖਺஋ܭࢉͰ͸਺஋ܭࢉʹΑΔޡࠩΛѻ͏. ਺஋ܭࢉʹΑΔޡࠩ a aେੴਐҰ. (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. ؙΊޡࠩ: ࣮਺ΛܭࢉػͰදݱՄೳͳ਺ʹஔ͖׵͑ͨ͜ͱʹΑΔޡࠩ ଧ੾Γޡࠩ: ແݶճ΍Δૢ࡞Λ༗ݶͰଧͪ੾ͬͨ͜ͱʹΑΔޡࠩ ཭ࢄԽޡࠩ (਺ཧϞσϧΛଟ߲ࣜ΍઴Խࣜʹม׵ͨ͠ࡍͷޡࠩ) ͨͩ͠, Ϟσϧޡࠩ (਺ཧϞσϧͦͷ΋ͷͷޡࠩ) ͸ѻΘͳ͍. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 5 / 33
6. ݚڀഎܠ q-ಛघؔ਺ q-ಛघؔ਺͸ύϥϝʔλ q ΛՃ͑ͨಛघؔ਺ͷ֦ு൛Ͱ͋Γ, q-ඍ෼΍ q-ੵ෼Λ ࢖͏ q-ղੳֶʹద߹͢ΔΑ͏ʹఆٛ͞ΕΔ (จࣈ q ͕͋Δ͚ͩͷؔ਺͸আ͘)2. 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 3 2ງాྑ೭, ౉ลܟҰ, ঙ࢘ढ़໌, ࡾொউٱ. (2004). ܈࿦ͷਐԽ, ୅਺ֶඦՊ I, ே૔ॻళ. 3Kac, V., Cheung, P. (2001). Quantum Calculus. Springer Science & Business Media. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 6 / 33
7. ݚڀഎܠ ݚڀͷҙٛ q-ಛघؔ਺͸ q-Painlev´ e ํఔࣜͳͲ༷ʑͳํఔࣜͷղͱͯ͠ݱΕΔ 4,5,6. q-ಛघؔ਺ͷੑ࣭ (ྵ఺, ෆಈ఺, ઴ۙతڍಈͳͲ) Λݚڀ͢Δʹ͸, q-ಛघ ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏͕ॏཁʹͳΓ͏Δ. طଘެࣜͷ਺஋తݕূʹ΋࢖͑Δ. 4Kajiwara, 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. 5Kemp, 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. 6Կ݈ࢤ, ᝛ࡾ࿠, ๺ࠜ༃࢙. (2007). ཭ࢄ֬཰աఔͱ q-௒زԿؔ਺. ೔ຊԠ༻਺ཧֶձ࿦จࢽ, 17(4), 463-468. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 7 / 33
8. q-gamma ؔ਺ q-gamma ؔ਺ q-gamma ؔ਺͸࣍ͷΑ͏ʹఆٛ͞ΕΔ 7. Γq (z) := (1 − q)1−z (q; q)∞ (qz; q)∞ , 0 < q < 1. (z; q)∞ ͸ q-Pochhammer ه߸ͱΑ͹Ε, ࣍ͷΑ͏ʹఆٛ͞ΕΔ. (z; q)n := ∏ n−1 k=0 (1 − zqk), (z; q)∞ := limn→∞ (z; q)n , ఆ͔ٛΒ, q-Pochhammer ه߸ (z; q)∞ Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖Ε͹, q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ͕Ͱ͖Δͱ͍͏͜ͱ͕෼͔Δ. ͔͜͜Β ͸ q-Pochhammer ه߸ (z; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏Λߟ͑Δ. 7Gasper, G., Rahman, M. (2004). Basic Hypergeometric Series. Cambridge University Press. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 8 / 33
9. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ q-Pochhammer ه߸ (z; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ q-Pochhammer ه߸ (z; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ͱͯ͠ (z; q)∞ Λަ୅ڃ਺ʹม׵ͯ͠ଧ੾ΓޡࠩΛධՁ͢Δํ๏ (z; q)∞ ͷۙࣅͱޡࠩ൒ܘΛར༻͢Δํ๏ Λ։ൃͨ͠. ަ୅ڃ਺Λ༻͍Δํ๏͔Βݟ͍ͯ͘. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 9 / 33
10. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ (z; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ (ަ୅ڃ਺Λ࢖༻) ఆٛ (q-ࢦ਺ؔ਺) eq (z) := ∞ ∑ n=0 zn (q; q)n , |z| < 1. ఆཧ (Euler) eq (z) = 1 (z; q)∞ ఆཧ (Karpelevich) 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 8. (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 ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 10 / 33
11. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ (z; q)∞ (q; q)∞ = 1/ ∞ ∑ n=0 (−1)nqn(n+1)/2 (q; q)n (1 − zqn) ΑΓ, (q; q)∞ ͱ ∑ ∞ n=0 (−1)nqn(n+1)/2 (q;q)n(1−zqn) Λਫ਼౓อূ෇͖਺஋ܭࢉ͢Ε͹ (z; q)∞ Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖Δ. (q; q)∞ ͱ ∑ ∞ n=0 (−1)nqn(n+1)/2 (q;q)n(1−zqn) ͷਫ਼౓อূ෇͖਺஋ܭࢉʹ͸ҎԼΛ࢖༻ͨ͠. 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਺ͷੑ࣭͕ద༻Ͱ͖ΔΑ͏ʹ (q; q)∞ Λมܗ͍ͯ͘͜͠ͱΛߟ͑Δ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 11 / 33
12. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ (q; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ (q; q)∞ ͸ Euler ؔ਺ͱΑ͹ΕΔ͜ͱ΋͋Γ, Euler ౳ʹΑͬͯ਺࿦ͷ෼໺Ͱݚڀ ͞Ε͍ͯΔؔ਺Ͱ͋Δ. Euler ʹΑͬͯҎԼͷΑ͏ͳมܗެ͕ࣜಋ͔Ε͍ͯΔ. ఆཧ (Euler ͷޒ֯਺ఆཧ) |q| < 1 ͷͱ͖, (q; q)∞ = ∞ ∏ n=1 (1 − qn) = 1 + ∞ ∑ n=1 (−1)n(1 + qn)qn(3n−1)/2 ͕੒Γཱͭ. ͜ͷެࣜΛ࢖ͬͯਫ਼౓อূ෇͖਺஋ܭࢉΛߦ͏͜ͱΛࢼΈ͕ͨ, ∑ ∞ n=1 (−1)n(1 + qn)qn(3n−1)/2 ʹ͍ͭͯ͸લड़ͷަ୅ڃ਺ͷੑ࣭Λద༻Ͱ͖ ͳ͍, ͭ·Γ, ڃ਺ͷத਎Λ dn ͱ͓͍ͨͱ͖ʹ dn ͷઈର஋͕୯ௐݮগ͠ͳ͍ͱ ͍͏͜ͱ͕෼͔ͬͨ. ͦ͜Ͱ Euler ͷޒ֯਺ఆཧͷผදݱΛ༻͍Δ͜ͱʹͨ͠. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 12 / 33
13. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ (q; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ Euler ͷޒ֯਺ఆཧʹ͸࣍ͷผදݱ͕͋Γ, ఆཧ A, B Λ༻͍ͯಋग़Ͱ͖Δ. 1 + ∞ ∑ n=1 (−1)n(1 + qn)qn(3n−1)/2 = ∞ ∑ n=0 (−1)n(1 − q2n+1)qn(3n+1)/2. ఆཧ A (Shanks ͷެࣜ) Shanks, D. (1951). A Short Proof of an Identity of Euler. Proceedings of the American Mathematical Society, 2(5), 747-749. 1 + N ∑ n=1 (−1)n(1 + qn)qn(3n−1)/2 = (q; q)N N ∑ n=0 (−1)nqNn+n(n+1)/2 (q; q)n ఆཧ B (Andrews, Merca) Andrews, G. E., Merca, M. (2012). The Truncated Pentagonal Number Theorem. Journal of Combinatorial Theory, Series A, 119(8), 1639-1643. N−1 ∑ n=0 (−1)n(1 − q2n+1)qn(3n+1)/2 = (q; q)N N−1 ∑ n=0 (−1)nqNn+n(n+1)/2 (q; q)n ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 13 / 33
14. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ (q; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ N → ∞ ͱ͢Δͱ, (ఆཧ A ͷӈล) = (q; q)N ( 1 − qN+1 (q;q)1 + · · · + (−1)N qN2+N(N+1)/2 (q;q)N ) → (q; q)∞ , (ఆཧ B ͷӈล) = (q; q)N ( 1 − qN+1 (q;q)1 + · · · + (−1)N−1qN(N−1)+N(N−1)/2 (q;q)N−1 ) → (q; q)∞ ͱͳΔ͜ͱ͔Β, limN→∞ (ఆཧ A ͷࠨล)= limN→∞ (ఆཧ A ͷӈล) = limN→∞ (ఆཧ B ͷӈล)= limN→∞ (ఆཧ B ͷࠨล). ∴ 1 + ∞ ∑ n=1 (−1)n(1 + qn)qn(3n−1)/2 = ∞ ∑ n=0 (−1)n(1 − q2n+1)qn(3n+1)/2. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 14 / 33
15. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ Euler ͷޒ֯਺ఆཧͷผදݱͰݱΕͨ ∞ ∑ n=0 (−1)n(1 − q2n+1)qn(3n+1)/2 ʹରͯ͠, લड़ͷަ୅ڃ਺ͷੑ࣭Λద༻Ͱ͖Δ. (q; q)∞ ͷଧ੾ΓޡࠩΛධՁͰ͖Δ. ∑ ∞ n=0 (−1)nqn(n+1)/2 (q;q)n(1−zqn) ʹ΋લड़ͷަ୅ڃ਺ͷੑ࣭Λద༻Ͱ͖ͯ, ଧ੾Γޡࠩ ΛධՁͰ͖Δ. (z; q)∞ (q; q)∞ = 1/ ∞ ∑ n=0 (−1)nqn(n+1)/2 (q; q)n (1 − zqn) . ຊख๏ͷ·ͱΊͱ՝୊ −1 < z < 1 ͷͱ͖ (z; q)∞ Λਫ਼౓อূ෇͖਺஋ܭࢉ͢Δ͜ͱ͕Ͱ͖ͨ. z ∈ C Ͱ΋ (z; q)∞ Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖ͳ͍͔ ? 2 ͭΊͷఏҊख๏Ͱ͸͜ͷ఺Λࠀ෰͢Δ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 15 / 33
16. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ q-Pochhammer ه߸ (z; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ (z; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉͰ͸ҎԼͷఆཧ 1, 2 ͕࢖͑Δ. ఆཧ 1 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 Ͱ͋Δͱ͖, ҎԼ͕੒Γཱͭ: (z;q)∞ (z;q)n = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn 1−q . ఆཧ 1 ͸ (z; q)n × (த৺ 1, ൒ܘ r(z; n) ͷۙ๣) ͱ͍͏ܗͷධՁͰ͋Δ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 16 / 33
17. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ ఆཧ 2 ͸ (z; q)∞ ͷۙࣅͱޡࠩΛ༩͑ΔఆཧͰ͋Δ. ఆཧ 2 a aGabutti, B., Allasia, G. (2008). Evaluation of q-Gamma Function and q-Analogues by Iterative Algorithms. Numerical Algorithms, 49(1), 159-168. z ∈ C, 0 < q < 1 ͱ͠, N ∈ N Λे෼େͳΔ਺ͱ͢ΔͱҎԼ͕੒Γཱͭ: (z; q)∞ − Tm−1,N (z) = (z; q)N ∞ ∑ j=0 dm+j (zqN )m+j, dk = qq2 · · · qk−1(1 − q)k (1 − q)(1 − q2) · · · (1 − qk) ≤ qk(k−1)/2, Tm,N (z) = (z; q)N m ∑ k=0 dk (zqN )k. ఆཧ 2 Ͱ |zqN | < 1 ΛԾఆ͢Ε͹ҎԼ͕੒Γཱͭ: |(z; q)∞ − Tm−1,N (z)| ≤ (z; q)N |zqN |mqm(m−1)/2 1 − |zqN | . ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 17 / 33
18. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ ਺஋࣮ݧ (ఆཧ 1, 2 ͷൺֱ) ఆཧ 1, 2 ʹΑΓ q-Pochhammer ه߸Λਫ਼౓อূ෇͖਺஋ܭࢉ͢ΔϓϩάϥϜΛ C++Ͱ࡞Γ, ܭࢉ݁ՌΛൺֱͨ͠. ࣮ݧͰ͸ C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉ ϥΠϒϥϦͰ͋Δ”kv ϥΠϒϥϦ”9 Λ࢖༻͍ͯ͠Δ. ࣮ݧ؀ڥ OS: Ubuntu14.04LTS, ίϯύΠϥʔ: gcc version 4.8.4 CPU: Intel Xeon(R) CPU E3-1241 v3 @ 3.50GHz × 8 ϝϞϦ: 15.6GB, kv ϥΠϒϥϦͷόʔδϣϯ: 0.4.42 ࣮ݧ݁Ռ (q = 0.1, z = 15) ਫ਼౓อূͷ݁Ռ (ఆཧ 1): [5.8509835632983984,5.8509835632985006] ਫ਼౓อূͷ݁Ռ (ఆཧ 2): [5.8509835632983975,5.8509835632985015] ͜ͷ৔߹Ͱ͸େ͖ͳҧ͍͸ͳ͍. 9ദ໦խӳ, kv - C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦ http://verifiedby.me/kv/index.html ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 18 / 33
19. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ ਺஋࣮ݧ (ఆཧ 1, 2 ͷൺֱ) ࠓ౓͸ q → 1 ͱͯ͠ΈΔ. ࣮ݧ݁Ռ (q = 0.9, z = 15) ਫ਼౓อূͷ݁Ռ (ఆཧ 1): [411.89219387077093,418.50918051346912] ਫ਼౓อূͷ݁Ռ (ఆཧ 2): [415.2006871920434,415.20068719219541] ఆཧ 1 Λ࢖͏ํ͕۠ؒ෯͕޿͕ͬͯ͠·͏. ͜Ε͸ఆཧ 1 Ͱ q → 1 ͱͨ͠ͱ͖, ఆཧͷԾఆ͕ຬͨ͞ΕΔΑ͏ʹ n → ∞ ͱͳͬͯܭࢉྔ͕૿͑ͯ͠·͏ͨΊͰ ͋Δ. q → 1 ͷͱ͖͸ఆཧ 2 Λ࢖ͬͯܭࢉΛߦ͏͜ͱʹ͢Δ. 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 Ͱ͋Δͱ͖, ҎԼ͕੒Γཱͭ: (z;q)∞ (z;q)n = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn 1−q . ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 19 / 33
20. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ q → 1, |z| → ∞ Ͱ͋Δ৔߹ͷରࡦ ఆཧ 2 ΑΓ, q → 1 ͱͨ͠ͱ͖ʹ q-Pochhammer ه߸Λਫ਼౓อূ෇͖਺஋ܭࢉ Ͱ͖ΔΑ͏ʹͳ͕ͬͨ, q → 1, |z| → ∞ ͱͨ͠ͱ͖ʹ q-gamma ؔ਺͕ਫ਼౓อ ূ෇͖਺஋ܭࢉͰ͖ΔΑ͏ʹͳͬͨΘ͚Ͱ͸ͳ͍. ࣮ݧ݁Ռ (z = −100 + 100i, q = 0.99) ਫ਼౓อূͷ݁Ռ: θϩআࢉൃੜ Γq (z) := (1 − q)1−z (q; q)∞ (qz; q)∞ , 0 < q < 1. ͜ͷ਺஋ྫͰ͸ 1 (qz;q)∞ Ͱθϩআࢉ͕ൃੜ͍ͯ͠Δ. ରࡦʹ͍ͭͯߟ͑Δ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 20 / 33
21. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ q → 1, |z| → ∞ Ͱ͋Δ৔߹ͷରࡦ q → 1, |z| → ∞ ͱͨ͠ͱ͖ʹ͸࣍Λ༻͍Δ. ఆཧ 3 (q-gamma ؔ਺ͷ৐ੵެࣜ, n ∈ N)a aGasper, G., Rahman, M. (2004). Basic Hypergeometric Series. Cambridge University Press. Γq (nz)Γr (1/n)Γr (2/n) · · · Γr ((n − 1)/n) = ( 1 − qn 1 − q )nz−1 Γr (z)Γr (z + 1/n) · · · Γr (z + (n − 1)/n), r = qn. qn ͱ͢Δ͜ͱͰ q Λ 1 ͔Βԕ͚ͯ͟, z/n ͱ͢Δ͜ͱͰ z Λখ͍ͯ͘͞͠Δ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 21 / 33
22. ݚڀ੒Ռ ৐ੵެࣜʹΑΔํ๏ ਺஋࣮ݧ (վྑલͱվྑޙͷൺֱ) ఆཧ 3 ʹΑΓ q-gamma ؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉ͢ΔϓϩάϥϜΛ C++Ͱ ࣗ࡞͠, վྑલͷܭࢉ݁ՌΛൺֱͨ͠. ࣮ݧ݁Ռ (z = −100 + 100i, q = 0.99, n = 20) վྑޙͷ݁Ռ: ([3.520226006915545×10−275,3.5245441358982552×10−275])+ ([-1.741771549163063×10−274,-1.7413396208989371×10−274])i վྑલͷ݁Ռ: θϩআࢉൃੜ Mathematica(ۙࣅ): 3.522378546643906×10−275-1.741556211443174×10−274i ܭࢉʹ੒ޭ͍ͯ͠Δ͜ͱ͔Β, վྑ͕੒ޭ͍ͯ͠Δͱ͍͑Δ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 22 / 33
23. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ ੵ෼ʹΑΔํ๏ ͜͜Ͱ͸ q-gamma ؔ਺ͷ࣋ͭ࣍ͷੵ෼දࣔ 10 Λѻ͏. 1 Γq (z) = sin(πz) π ∫ ∞ 0 t−zdt (−t(1 − q); q)∞ . ੵ෼ʹΑΔ q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉΛߦ͏લʹ, ઌߦݚڀͰ͋Δ gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏ʹ͍ͭͯݟ͍ͯ͘. 10Ismail, M. E. (1981). The Basic Bessel Functions and Polynomials. SIAM Journal on Mathematical Analysis, 12(3), 454-468. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 23 / 33
24. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ ઌߦݚڀ (ੵ෼ʹΑΔ gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ) ੵ෼ʹΑΔ gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ͸࣍ͷྲྀΕͰߦΘΕͨ. Γ(z) = ∫ ∞ 0 xz−1e−x dx. ੵ෼ʹΑΔ gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ a aڮຊਸر, ദ໦խӳ, ෳૉ Gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ, ೔ຊԠ༻਺ཧֶձ೥ ձ, ෢ଂ໺େֶ, 2017 ೥ 9 ݄ 6-8 ೔ ੵ෼දࣔΛ࣮෦ͱڏ෦ʹ෼ׂ͢Δ. ↓ ੵ෼۠ؒΛ [0, 1], [1, T ], [T, ∞] ͷ 3 ͭʹ෼͚ͯੵ෼ܭࢉΛߦ͏ (T > 1). ↓ [0, 1] Ͱ͸த਎Λڃ਺ʹ௚߲ͯ͠ผੵ෼͠, [1, T ] Ͱ͸ kv ϥΠϒϥϦͷਫ਼౓อূ ෇͖਺஋ੵ෼ύοέʔδΛ࢖͏. ↓ [T, ∞] Ͱ͸ݫີʹੵ෼஋ͷٻΊΒΕΔؔ਺Ͱඃੵ෼ؔ਺ΛධՁ͢Δ. ͜ͷํ๏Λࢀߟʹ, ੵ෼ʹΑΔ q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉΛߦ͏. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 24 / 33
25. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ ੵ෼ʹΑΔํ๏ q-gamma ؔ਺ͷੵ෼දࣔΛ࣮෦ͱڏ෦ʹ෼͚Δͱ࣍ͷΑ͏ʹͳΔ (z = x + iy). 1 Γq (z) = sin(πz) π (∫ ∞ 0 t−x cos(y log t)dt (−t(1 − q); q)∞ − i ∫ ∞ 0 t−x sin(y log t)dt (−t(1 − q); q)∞ ) . ࣮෦ͱڏ෦Ͱੵ෼۠ؒΛ [0, 1], [1, T ], [T, ∞] ͷ 3 ͭʹ෼͚ͯੵ෼ܭࢉΛߦ͏ (T > 1). ͔͜͜Β͸࣮෦ͷੵ෼ܭࢉʹ͍ͭͯݟ͍ͯ͘ (ڏ෦΋ಉ༷). ·ͨ, 1 ≤ x ≤ 2 ΛԾఆ͢Δ. ؔ਺౳ࣜ: Γq (z + 1) = 1 − qz 1 − q Γq (z) ΑΓ, x ͕ଞͷൣғʹ͋Δ৔߹Ͱ΋ਫ਼౓อূ෇͖਺஋ܭࢉͰ͖ΔͨΊ 1 ≤ x ≤ 2 ΛԾఆͯ͜͠ͷൣғͰͷධՁΛߟ͑Ε͹ྑ͍. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 25 / 33
26. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ ੵ෼ʹΑΔํ๏ [0, 1] Ͱ͸, ิॿతʹҎԼͷؔ਺Λ༻͍Δ. ఆٛ (q-ࢦ਺ؔ਺) Andrews, G. E., Askey, R., Roy, R. (1999). Special Functions, Cambridge University Press. eq (z) := ∞ ∑ n=0 zn (q; q)n , |z| < 1. ఆཧ 4 (Euler) Andrews, G. E., Askey, R., Roy, R. (1999). Special Functions, Cambridge University Press. eq (z) = 1 (z; q)∞ ఆཧ 4 Λ࢖ͬͯඃੵ෼ؔ਺Λมܗ͠, ߲ผੵ෼Λߦͳ͏. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 26 / 33
27. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ ੵ෼ʹΑΔํ๏ ఆཧ 4 Λ࢖߲ͬͯผੵ෼, ෦෼ੵ෼͢Δͱ, ∫ 1 0 t−x cos(y log t) (−t(1 − q); q)∞ dt = ∞ ∑ n=0 (−(1 − q))n (q; q)n n − x + 1 (n − x + 1)2 + y2 ͱͳΔ. S(n) = (−(1−q))n (q;q)n n−x+1 (n−x+1)2+y2 ͱ͓͘ͱެൺ͸ n ≥ N ͷͱ͖, S(n + 1) S(n) = 1 − q 1 − qn+1 n − x + 2 n − x + 1 (n − x + 1)2 + y2 (n − x + 2)2 + y2 ≤ 1 − q 1 − qN+1 ( 1 + 1 N − x + 1 ) =: D ͱͳΔ. D < 1 ͷͱ͖ଧͪ੾ΓޡࠩΛҎԼͷΑ͏ʹධՁͰ͖Δ: ∞ ∑ n=N S(n) ≤ (ॳ߲ |S(N)|, ެൺ D ͷ౳ൺ਺ྻͷ࿨) = |S(N)| 1 − D . ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 27 / 33
28. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ [1, T ] Ͱੵ෼͢Δࡍ͸ඃੵ෼ؔ਺: t−x cos(y log t) (−t(1 − q); q)∞ ʹ͋Δ q-Pochhammer ه߸Λੵ෼͠΍͍͢ܗʹมܗ͢Δ. มܗʹ͸ఆཧ 5 Λ࢖͏. ఆཧ 5 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 Ͱ͋Δͱ͖, ҎԼ͕੒Γཱͭ: (z;q)n (z;q)∞ = 1 + r(z; n), |r(z; n)| ≤ 2|z|qn 1−q . ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 28 / 33
29. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ ੵ෼ʹΑΔํ๏ ఆཧ 5 Λ࢖͏ͱඃੵ෼ؔ਺͸ t−x cos(y log t) (−t(1 − q); q)∞ ∈ t−x cos(y log t) (−t(1 − q); q)n [ 1 ± 2|t(1 − q)|qn 1 − q ] ͱมܗͰ͖Δ. มܗޙʹݱΕΔ n ∈ N ͸ 0 < 2|t(1 − q)|qn 1 − q < 1 2 (ఆཧ 5 ͷԾఆ) Λຬͨ͢Α͏ʹऔΔ. ·ͨ, [a ± b] := [a − b, a + b] ͱఆΊΔ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 29 / 33
30. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ ੵ෼ʹΑΔํ๏ 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͸ 90 ݸʹ෼ׂ) ⇓ ඃੵ෼ؔ਺ f ʹରͯ͠৒༨߲෇͖ Taylor ల։Λߦ͏ ⇓ ֤۠෼Ͱ f ͷ૾Λ܎਺͕۠ؒͰ͋Δଟ߲ࣜͱͯ͠ಘΔ (࣮ݧͰ͸ 90 ࣍ʹࢦఆ) ⇓ ֤۠෼ͰಘΒΕͨଟ߲ࣜΛෆఆੵ෼ͯ͠ݪ࢝ؔ਺ΛಘΔ ⇓ ֤۠෼Ͱ۠ؒ୺ͷ஋Λ୅ೖͯ͠ఆੵ෼ͷ஋Λ۠ؒͱͯ͠ಘΔ ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 30 / 33
31. ݚڀ੒Ռ ੵ෼ʹΑΔํ๏ [T, ∞] Ͱͷඃੵ෼ؔ਺ͷධՁ͸ x ∈ [1, 2] ͱఆཧ 5 ΑΓ࣍ͷΑ͏ʹͳΔ: t−x cos(y log t) (−t(1 − q); q)∞ ∈ [ − 1 − 2tqn/(1 − q) t2(1 + t(1 − q))n , 1 + 2tqn/(1 − q) t(1 + t(1 − q)qn−1)n ] . ্ݶͱԼݶΛͦΕͧΕෆఆੵ෼͢Δͱ 11 ࣍ͷΑ͏ʹͳΔ: ∫ 1 + 2tqn/(1 − q) t(1 + t(1 − q)qn−1)n dt = ( 1 − (q − 1)tqn−1 ) −n ×  2 ( tqn+1 − tqn − q ) (n − 1)(q − 1)2 − ( q1−n t−qt + 1 ) n 2F1 ( n, n; n + 1; q1−n (q−1)t ) n   , − ∫ 1 − 2tqn/(1 − q) t2(1 + t(1 − q))n dt = (−qt + t + 1)−n ( 1 t − qt + 1 )n × ( 2(n + 1)tqn 2F1 ( n, n; n + 1; 1 (q − 1)t ) +n(q − 1)2F1 ( n, n + 1; n + 2; 1 (q − 1)t )) /{n(n + 1)(q − 1)t}. 11ෆఆੵ෼ʹ͸ Mathematica ͷ Integrate Λ༻͍ͨ. ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 31 / 33
32. ਺஋࣮ݧ (ఏҊख๏ͱ Mathematica ͱͷൺֱ) ਺஋࣮ݧ (ੵ෼) q-gamma ؔ਺Λੵ෼ʹΑͬͯਫ਼౓อূ෇͖਺஋ܭࢉ͢ΔϓϩάϥϜΛ C++Ͱࣗ ࡞͠, ਺஋࣮ݧΛߦͬͨ. ࣮ݧʹ͸ C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦ Ͱ͋Δ”kv ϥΠϒϥϦ”Λ࢖༻͍ͯ͠Δ. ࣮ݧ؀ڥ OS: Ubuntu14.04LTS, ίϯύΠϥʔ: gcc version 4.8.4 CPU: Intel Xeon(R) CPU E3-1241 v3 @ 3.50GHz × 8 ϝϞϦ: 15.6GB, kv ϥΠϒϥϦͷόʔδϣϯ: 0.4.42 ࣮ݧ݁Ռ (z = 1.2 + i, q = 0.1) ਫ਼౓อূͷ݁Ռ (ੵ෼):([0.48702750125220128,2.2700419350688224])+ ([-0.46939808387927973,1.1606835821327298])i Mathematica (ۙࣅ): 0.865915330766597+0.04780558270549618i ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 32 / 33
33. ຊݚڀͷ·ͱΊͱࠓޙͷ՝୊ ຊݚڀͷ·ͱΊͱࠓޙͷ՝୊ ຊݚڀͷ·ͱΊ q-gamma ؔ਺Λهड़͢Δ q-Pochhammer ه߸Λਫ਼౓อূ෇͖਺஋ܭࢉͨ͠. ৐ੵެࣜͱੵ෼ʹΑͬͯ q-gamma ؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉͨ͠. ࠓޙͷ՝୊ ੵ෼ʹΑΔํ๏Ͱ͸۠ؒ෯ͷ޿͕ΓΛ཈͑Δ͜ͱ͕Ͱ͖ͳ͔ͬͨ. ඃੵ෼ؔ਺ͷධՁΛ͞Βʹݫ͘͢͠Δ͜ͱͰվྑͰ͖ͳ͍͔ ? ۚઘେհ (ૣҴాେֶ, M1), ؙ໺݈Ұ (ૣҴాେֶ) q-gamma ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 2017 ೥ 12 ݄ 14-16 ೔ 33 / 33