q-Bessel 関数の精度保証付き数値計算・零点探索

Verified computation of q-Bessel functions and their zeros.

Transcript

1. q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) ˏദ໦ݚڀࣨ, ૣҴాେֶ਺ֶԠ༻਺ཧઐ߈ 0͜ͷൃද͸ୈ̐ճֶੜݚڀൃදձ (ˏஜ೾େֶ, 2019 ೥ 3 ݄ 3 ೔) ͰͷߨԋΛՃචͨ͠΋ͷͰ͢. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 1 / 54
2. ຊൃදͷྲྀΕ 1 ݚڀഎܠ ਫ਼౓อূ෇͖਺஋ܭࢉ q-ಛघؔ਺ 2 ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ q-Bessel ؔ਺ͷྵ఺୳ࡧ 3 ·ͱΊͱ՝୊ 4 ଞʹम࿦౳Ͱ΍ͬͨ͜ͱ 5 ࢀߟจݙ ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 2 / 54
3. ݚڀഎܠ ݚڀഎܠ Մੵ෼ܥ౳ͰݱΕΔ q-ಛघؔ਺ (q-gamma, q-Airy, q-Bessel, modified q-Bessel, q-௚ަଟ߲ࣜ etc) ʹ͍ͭͯ͸༷ʑͳݚڀ͕ͳ͞Ε͍ͯΔ (Gasper-Rahman, Andrews-Askey-Roy, Ernst, Exton, Ismail, Kac-Cheung, Koornwinder, Koelink, Koekoek-Lesky-Swarttouw, DLMF etc). ਫ਼౓อূ෇͖਺஋ܭࢉʹΑΔ q-ಛघؔ਺ͷݚڀΛ໨ࢦ͢. ࠓ·Ͱ Jν (x) ౳, ༷ʑͳಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏͕։ൃ͞Ε͕ͨ (Arb, INTLAB, kv, Yamamoto-Matsuda, Yamanaka-Okayama-Oishi etc), q-ಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏͸·ͩͳ͍ 1,2,3. q-ಛघؔ਺ (ಛघؔ਺ͷ q-ྨࣅ) Weisstein, Eric W. ”q-Analog.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/q-Analog.html ύϥϝʔλ q ΛՃ͑ΔҰൠԽ (͜ͷ q ͸ base ͱݺ͹ΕΔ) q → 1 ͱͨ͠ͱ͖ݩʹ໭Δ 1Mathematica ͳΒ QHypergeometricPFQ, QGamma, QPochhammer Λ࢖ͬͯۙࣅ஋ܭࢉͰ͖Δ. 2ޙड़͢Δ͕ q-Pochhammer ه߸ͷਫ਼౓อূ෇͖਺஋ܭࢉʹؔ͢Δઌߦݚڀ͸͋Δ. 3q-ಛघؔ਺͸ xy = qyx(ྔࢠ܈), ͭ·ΓܭࢉػͰදݱͰ͖ͳ͍ੈքͰਅՁΛൃش͢Δ. Αͬͯ, ʮq-ಛघؔ਺ΛܭࢉػͰѻ͏͜ͱ͸ແҙຯͩʯͱ͍͏൷൑΋੒ཱ͠͏Δ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 3 / 54
4. ݚڀഎܠ ਫ਼౓อূ෇͖਺஋ܭࢉ ਫ਼౓อূ෇͖਺஋ܭࢉͰѻ͏ޡࠩ ਫ਼౓อূ෇͖਺஋ܭࢉ ۙࣅ஋ͷܭࢉΛ͢Δͱಉ࣌ʹܭࢉ݁Ռͷ (਺ֶతʹ) ݫີͳޡࠩධՁ΋ߦ͏਺஋ ܭࢉͷ͜ͱ. ਅ஋ΛؚΉ۠ؒΛ݁Ռͱͯ͠ग़ྗ͢Δ. ਫ਼౓อূ෇͖਺஋ܭࢉͰ͸਺஋ܭࢉʹΑΔޡࠩΛѻ͏. ਺஋ܭࢉʹΑΔޡࠩ େੴਐҰ et al. (2018). ਫ਼౓อূ෇͖਺஋ܭࢉͷجૅ, ίϩφࣾ. େੴਐҰ (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. Tucker, W. (2011). Validated numerics: a short introduction to rigorous computations. Princeton University Press. ؙΊޡࠩ: ࣮਺ΛܭࢉػͰදݱՄೳͳ਺ʹஔ͖׵͑ͨ͜ͱʹΑΔޡࠩ ଧ੾Γޡࠩ: ແݶճ΍Δૢ࡞Λ༗ݶͰଧͪ੾ͬͨ͜ͱʹΑΔޡࠩ ཭ࢄԽޡࠩ: ਺ཧϞσϧΛଟ߲ࣜ΍઴Խࣜʹม׵ͨ͠ࡍͷޡࠩ ͨͩ͠, Ϟσϧޡࠩ (਺ཧϞσϧͦͷ΋ͷͷޡࠩ) ͸ѻΘͳ͍. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 4 / 54
5. ݚڀഎܠ ਫ਼౓อূ෇͖਺஋ܭࢉ ۠ؒԋࢉ ਫ਼౓อূ෇͖਺஋ܭࢉͰ͸۠ؒԋࢉʹΑΓ਺஋ܭࢉͰੜ͡ΔޡࠩΛ೺Ѳ͍ͯ͠Δ. Definition (۠ؒԋࢉ) ۠ؒԋࢉΛߦ͏ࡍ͸਺Λด۠ؒʹஔ͖׵͑ͯԼهͷϧʔϧʹै͍ܭࢉ͍ͯ͠Δ. (¯ x ্͕ݶ, x ͕Լݶ,[x] = [x, ¯ x] ͱ͢Δ) [x] + [y] := [x + y, ¯ x + ¯ y], [x] − [y] := [x − ¯ y, ¯ x − y] ৐ࢉ: [x] × [y] := [min(xy, x¯ y, y¯ x, ¯ xy), max(xy, x¯ y, y¯ x, ¯ xy)] আࢉɹ (ͨͩ͠ [y] ͸ 0 Λؚ·ͳ͍۠ؒͱ͢Δ): [x]/[y] := [min(x/y, x/¯ y, ¯ x/y, ¯ x/¯ y), max(x/y, x/¯ y, ¯ x/y, ¯ x/¯ y)] େੴਐҰ et al. (2018). ਫ਼౓อূ෇͖਺஋ܭࢉͷجૅ, ίϩφࣾ. େੴਐҰ (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. Tucker, W. (2011). Validated numerics: a short introduction to rigorous computations. Princeton University Press. Mayer, G. (2017). Interval analysis: and automatic result verification (Vol. 65). Walter de Gruyter GmbH & Co KG. Alefeld, G., & Herzberger, J. (2012). Introduction to interval computation. Academic press. Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to interval analysis (Vol. 110). SIAM. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 5 / 54
6. ݚڀഎܠ q-ಛघؔ਺ q-ྨࣅ ࣗવ਺ n ʹରͯ͠ 1−qn 1−q = 1 + q + q2 + · · · + qn−1 Λߟ͑Δ࣌, lim q→1 1 − qn 1 − q = lim q→1 (1 + q + q2 + · · · + qn−1) = n ͱͳΔͷͰ, 1−qn 1−q ͸ࣗવ਺ n ͷ q ྨࣅͰ͋Δ. Αͬͯ, ࣗવ਺ n ͷ q ྨࣅΛ [n]q := 1−qn 1−q ͱॻ͘. ͜ΕΛ༻͍ͯ, q-֊৐ [n] q ! := [n] q [n − 1] q · · · [2] q [1] q ΍, q-ೋ߲܎਺ [ n k ] q := [n] q [n−1] q ···[n−k] q [k] q ΛఆΊΒΕΔ. ͜͜Ͱ, n, N ∈ N ʹରͯ͠, [ N + n n ] q Λ q ʹ͍ͭͯల։ͯ͠, [ N + n n ] q = ∑ k c(n, N, k)qk ͱදΘ͢ͱ, ܎਺ c(n, N, k) ͸, ”k ∈ N Λ ߴʑn ݸͷ N ҎԼͷࣗવ਺ͷ࿨Ͱද͢৔߹ͷ਺” ͱ͍͏૊Έ߹Θͤ࿦తͳҙຯ ߹͍Λ࣋ͪ, q-ೋ߲܎਺͸͋Δछͷ฼ؔ਺ʹͳ͍ͬͯΔ 4. q ྨࣅ͸૊Έ߹Θͤ࿦ ͷ෼໺Ͱ΋ݱΕΔͷͰ͋Δ. 4Andrews, G., Eriksson, K. (2004). Integer Partitions, Cambridge University Press. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 6 / 54
7. ݚڀഎܠ q-ಛघؔ਺ q-ྨࣅ q-֊৐΍ q-ೋ߲܎਺Λ༻͍Δͱ͍Ζ͍Ζͳؔ਺ͷ q ྨࣅΛߟ͑Δ͜ͱ͕Ͱ͖Δ. ྫ͑͹, (Euler ͷ) q-ࢦ਺ؔ਺ 5,6: eq (z) := ∞ ∑ k=0 zk [k]q ! ΍ (Koekoek-Swarttouw ͷ) q-ࡾ֯ؔ਺ 7: cosq (z) := ∞ ∑ k=0 (−1)k z2k [2k]q ! , sinq (z) := ∞ ∑ k=0 (−1)k z2k+1 [2k + 1]q ! ͕͋Δ. q-ॳ౳ؔ਺Ҏ֎ʹ΋, q-ಛघؔ਺΋ߟ͑Δ͜ͱ͕Ͱ͖Δ. 5ࠇ໦ݰ, q-exponential, https://genkuroki.github.io/documents/. 6Euler ͷ q-ࢦ਺ؔ਺Ҏ֎ʹ΋༷ʑͳ q-ࢦ਺ؔ਺͕ Atakishiyev-Suslov (1992), Nelson-Gartley (1994), Ismail-Zhang (1994), Suslov (2003) ౳ʹΑͬͯݚڀ͞Ε͍ͯΔ. 7Gosper (2001) Ͱ͸શ͘ҟͳΔ q-ࡾ֯ؔ਺͕ݚڀ͞Ε͍ͯΔ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 7 / 54
8. ݚڀഎܠ q-ಛघؔ਺ q-ಛघؔ਺ q-ಛघؔ਺͸ύϥϝʔλ q ΛՃ͑ͨಛघؔ਺ͷ֦ு൛Ͱ͋Γ, q-ඍ෼΍ q-ੵ෼Λ ࢖͏ q-ղੳֶͱ૬ੑ͕͍͍Α͏ʹఆٛ͞ΕΔ (จࣈ q Λ࢖͏͚ͩͷؔ਺͸আ͘). q-ඍ෼ (q-ࠩ෼, Jackson ඍ෼) Dqf(x) := f(x) − f(qx) x(1 − q) q-ੵ෼ (Jackson ੵ෼) ∫ 1 0 f(t)dqt := (1 − q) ∞ ∑ n=0 f(qn)qn q-ಛघؔ਺͸ Euler ʹΑΔࣗવ਺ͷ෼ׂʹؔ͢ΔݚڀͳͲͰॳΊͯݱΕ, 19 ੈل ͔Β Jacobi ΒʹΑͬͯ q-ղੳֶͷ؍఺͔Βݚڀ͞ΕΔΑ͏ʹͳͬͨ. ͜ΕΒͷ ࣌୅ʹ͸ q-ಛघؔ਺ͷ਺ֶతഎܠ͸ෆ໌Ͱ͕͋ͬͨ, 1980 ೥୅ʹ Drinfeld-Jimbo ʹΑͬͯղ໌͞Εͨ. q-ղੳֶ͸༷ʑͳؔ਺ͷ q ྨࣅͷੑ࣭Λཧղ͢Δҝͷಓ۩ Ͱ͋Δ͚ͩͰͳ͘, ”q ͷੈք” Ҏ֎ͷ৔ॴʢྫ͑͹਺ཧ෺ཧʣͰ΋සൟʹొ৔͠ ༗ӹͳ݁ՌΛ΋ͨΒͯ͘͠ΕΔͷͰ͋Δ. ງాྑ೭, ౉ลܟҰ, ঙ࢘ढ़໌, ࡾொউٱ (2004). ܈࿦ͷਐԽ, ୅਺ֶඦՊ I, ே૔ॻళ. ੢ᖒಓ஌. (2015). q-ΨϯϚؔ਺ (ಛू ΨϯϚؔ਺ͱ͸Կ͔). ਺ֶηϛφʔ, 54(10), 28-33. ্໺تࡾ༤. (1997). q-ղੳֶͱྔࢠ܈ (ϑΥʔϥϜ: ݱ୅਺ֶͷ෩ܠ/q-ղੳֶͷϧωαϯε). ਺ֶͷͨͷ͠Έ, (2), 32-46. S. Ikebe, Graphics Library of Special Functions (ಛघؔ਺άϥϑΟ οΫεϥΠϒϥϦ). ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 8 / 54
9. ݚڀഎܠ q-ಛघؔ਺ ݚڀͷҙٛ q-ಛघؔ਺͸ q-Painlev´ e ํఔࣜͳͲ༷ʑͳํఔࣜͷղͱͯ͠ݱΕΔ 8,9 q-ಛघؔ਺ͷੑ࣭ (ྵ఺, ෆಈ఺, ઴ۙతڍಈͳͲ) Λݚڀ͢Δʹ͸, q-ಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏͕ॏཁʹͳΓ͏Δ. 8Kajiwara, 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. 9Kemp, A. (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. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 9 / 54
10. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ࠓճ͸ 2 छྨͷ q-Bessel ؔ਺Λ࢖࣮ͬͯݧ͢Δ. (ͨͩ͠ |q| < 1, ν ∈ C) J(2) ν (x; q) := (qν+1; q)∞ (q; q)∞ ( x 2 )ν 0 ϕ1 ( −; qν+1, q, − qν+1x2 4 ) , x ∈ C, J(3) ν (x; q) := (qν+1; q)∞ (q; q)∞ xν 1 ϕ1 (0; qν+1, q, qx2), x ∈ C, (a; q)n := n−1 ∏ k=0 (1 − aqk), (a; q)∞ := lim n→∞ (a; q)n (q-Pochhammer ه߸ 10 ,11 ), rϕs(α1, · · · , αr; β1, · · · , βs; q, z) := ∞ ∑ n=0 r ∏ i=1 (αi; q)n [ (−1)nq(n 2 ) ] 1+s−r zn s ∏ j=1 (βj; q)n(q; q)n . ্͔Βॱʹ Jackson ͷୈ 2 छ q-Bessel ؔ਺, Hahn-Exton ͷ q-Bessel ؔ਺ͱ Α͹Ε͍ͯΔ. ͜ΕΒ͸ q-Painlev´ e III ܕํఔࣜͷಛघղΛهड़͢Δ (Kajiwara-Ohta-Satsuma (1995), Kajiwara-Masuda-Noumi-Ohta-Yamada (2004)). 10Ұൠʹ q-ಛघؔ਺͸ (z; q)∞ ͱ rϕs Λ༻͍ͯఆٛ͞ΕΔ. Pochhammer ه߸ͱҟͳΓ, q-Pochhammer ه߸͸ແݶͷ৔߹΋ఆٛͰ͖Δ. ͜ͷ͜ͱ͕ q-ಛघؔ਺ͷੈքΛ๛য়ʹ͍ͯ͠Δ. 11(z; q)∞ ʹࣅͨؔ਺ͱͯ͠ Schottky Klein prime function ͱ͍͏ͷ͕͋Δͦ͏Ͱ͢. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 10 / 54
11. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 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 2 ) ]l zn ∏ s j=1 (βj ; q)n (q; q)n ͷଧ੾ΓޡࠩΛධՁͨ͠. ޡࠩධՁ (r ≤ s ͷͱ͖) T (n) := ∏ r i=1 (αi;q)n [ (−1)nq (n 2 ) ] l zn ∏ s j=1 (βj ;q)n(q;q)n ͱ͓͘ͱ, 0 < q < 1, |βj | ≤ q−N ͷ ͱ͖, ∞ ∑ 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. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 11 / 54
12. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ 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 2 ) ]l zn ∏ s j=1 (βj ; q)n (q; q)n ͷଧ੾ΓޡࠩΛධՁͨ͠. ޡࠩධՁ (r = s + 1 ͷͱ͖) 0 < q < 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| . ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 12 / 54
13. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ূ໌ʹ͸࣍ͷิ୊Λ༻͍Δ. Lemma n, N ∈ Z≥0 , n ≥ N, 0 < q < 1, c ∈ C ͱ͢Δ. |c| ≤ q−N ͷͱ͖, qn |1 − cqn| ≤ qN |1 − cqN | ͕੒Γཱͭ. 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|. ∴ qn |1−cqn| ≤ qN |1−cqN | . ิ୊͕ࣔ͞Εͨ. 2 ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 13 / 54
14. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ޡࠩධՁͷূ໌ r ≤ s ͷͱ͖, 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 ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 14 / 54
15. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ޡࠩධՁͷূ໌ r = s + 1 ͷͱ͖, 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 ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 15 / 54
16. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ (z; q)∞ ͷਫ਼౓อূ෇͖਺஋ܭࢉʹ͸࣍ͷఆཧΛ࢖͏ 12,13,14. Theorem z ∈ C, 0 < q < 1 ͱ͢Δ. ਖ਼ͷ੔਺ m ʹରͯ͠ 0 < |z|qm 1−q < 1 2 Ͱ͋Δͱ͖, (z; q)∞ (z; q)m = 1 + r(z; m), |r(z; m)| ≤ 2|z|qm 1 − q ͕੒Γཱͭ. Zhang, R. (2008). Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series, Advances in Mathematics 217, 1588-1613. ͪͳΈʹ (A; q)∞ , A ∈ Cn×n Ͱ΋্ͷఆཧͱྨࣅͨ͠ओு͕੒Γཱͭ (ৄ͘͠ ͸ࢲͷम࿦Λࢀর). 12͜ͷํ๏ͩͱ q → 1 ·ͨ͸ |z| → ∞ ͷ࣌͸ܭࢉࠔ೉Ͱ͋Δ. 13ࣅͨΑ͏ͳެ͕ࣜ Zhang, R. (2008). On asymptotics of q-Gamma functions. Journal of Mathematical Analysis and Applications, 339(2), 1313-1321. ͷ Lemma 1.1 ʹ͋Δ. 14(q; q)∞ ʹରͯ͠͸͜ͷൃදͰ঺հ͢Δํ๏ͷଞʹ΋, ʮEuler ͷޒ֯਺ఆཧʯͷผදݱͱަ୅ ڃ਺ͷੑ࣭Λ૊Έ߹Θͤͨํ๏΋ߟ͑ΒΕΔ. ͜ͷख๏͸ 2017 ೥ 3 ݄ͷ೔ຊԠ༻਺ཧֶձݚڀ෦ձ ࿈߹ൃදձͰใࠂ͍͍ͤͯͨͩͨ͞. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 16 / 54
17. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ࣍ͷఆཧΛ࢖͑͹ q → 1 ͷͱ͖΋ (z; q)∞ Λ͋Δఔ౓ܭࢉͰ͖Δ. Theorem (Gabutti-Allasia, (z; q)∞ ͷۙࣅ) 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. Gabutti, B., Allasia, G. (2008). Evaluation of q-Gamma Function and q-Analogues by Iterative Algorithms. Numerical Algorithms, 49(1), 159-168. ܥ: ͜ͷఆཧͰ |zqN | < 1 ΛԾఆ͢Ε͹ҎԼ͕੒Γཱͭ: |(z; q)∞ − Tm−1,N (z)| ≤ (z; q)N |zqN |mqm(m−1)/2 1 − |zqN | . ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 17 / 54
18. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ x ͷઈର஋͕େ͖͍࣌ͷରࡦ q-௒زԿؔ਺ͱ q-Pochhammer ه߸ͷਫ਼౓อূ෇͖਺஋ܭࢉ͕Ͱ͖ͨ͜ͱ͔Β, 2 ͭͷ q-Bessel ؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖Δ. ͕ͩ, ͜ͷ··ͩͱ x ͷ ઈର஋͕େ͖͍ͱ͖ʹ݁Ռͷ۠ؒ෯͕ inf ͱͳΓ͏Δ. ࣮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: 0.4.41 ࣮ݧ݁Ռ (Jackson ͷୈ 2 छ q-Bessel ؔ਺, Hahn-Exton ͷ q-Bessel ؔ਺) ਺஋ྫ: q = 0.1, x = 40000, ν = 4.5 ਫ਼౓อূ෇͖਺஋ܭࢉͷ݁Ռ (۠ؒ): [-inf, inf] ڃ਺ͷܭࢉ࣌ʹ xn ͷ෦෼͕ inf ͱͳΔ͔ΒͰ͋Δ. ڃ਺ʹ xn ͕ͳ͍ผදݱΛ ༻͍Δ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 18 / 54
19. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷผදݱ Theorem (Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷผදݱ) ެࣜ: (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. Koelink, H. (1993). Hansen-Lommel Orthogonality Relations for Jackson’s q-Bessel Functions. Journal of Mathematical Analysis and Applications, 175, 425-437. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 19 / 54
20. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷผදݱ ྫ͑͹ ν < 0 ͷͱ͖͸લϖʔδͷެࣜΛ࢖ͬͯ΋ܭࢉ͕͏·͍͔͘ͳ͍͜ͱ͕ ͋Δ. ͜͏͍͏৔߹͸࣍Λ࢖͏ 15: 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 ) . 15Chen, Y., Ismail, M., Muttalib, K. (1994). Asymptotics of Basic Bessel Functions and q-Laguerre Polynomials, Journal of Computational and Applied Mathematics, 54, 263-272. ͜ͷ࿦จʹ͸ q-Laguerre ଟ߲ࣜͷผදݱ΋ܝࡌ͞Ε͍ͯͯ, q-Laguerre ଟ߲ࣜͷਫ਼౓อূ෇͖਺஋ ܭࢉʹॏๅ͍ͯ͠Δ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 20 / 54
21. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ Hahn-Exton ͷ q-Bessel ؔ਺ͷผදݱ Lemma (Koornwinder-Swarttouw) (w; q)∞ 1 ϕ1 (0; w; q, z) = (z; q)∞ 1 ϕ1 (0; z; q, w) Koornwinder, T., Swarttouw, R. (1992). On q-Analogues of the Fourier and Hankel Transforms. Transactions of the American Mathematical Society, 333, 445-461. Theorem (Hahn-Exton ͷ q-Bessel ؔ਺ͷผදݱ) ิ୊ΑΓ, Hahn-Exton ͷ q-Bessel ؔ਺͸࣍ͷผදݱΛ࣋ͭ: J(3) ν (x; q) = xν (x2q; q)∞ (q; q)∞ 1 ϕ1 (0; x2q; q, qν+1). ڃ਺ͷத਎ʹ xn Λ࣋ͨͳ͍Α͏ͳผදݱͰ͋Δ. Daalhuis, A. (1994). Asymptotic Expansions for q-Gamma, q-Exponential, and q-Bessel functions. Journal of Mathematical Analysis and Applications, 186, 896-913. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 21 / 54
22. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ਺஋࣮ݧ (վྑલͱվྑޙͷൺֱ) Jackson ͷୈ 2 छ q-Bessel ؔ਺ͱ Hahn-Exton ͷ q-Bessel ؔ਺ʹ͍ͭͯ਺஋࣮ݧ Λߦ͍, վྑલͱվྑޙͷൺֱΛߦͬͨ. ࣮ݧͰ͸ kv ϥΠϒϥϦ 16 Λ࢖͍ͬͯΔ. 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: 0.4.41 ਺஋ྫ: q = 0.1, x = 40000, ν = 4.5 J(2) ν (x; q):[3.6310367829349115,3.6310367829357793]×1023 J(3) ν (x; q):[-1.1387663357821531,-1.1387663357818429]×1058 ൃࢄ͕๷͛ͨ͜ͱ͔Β, վྑ͕੒ޭ͍ͯ͠Δͱݴ͑Δ. Ҏ্ͷ४උͷԼͰ, q-Bessel ؔ਺ͷྵ఺ଘࡏൣғΛ਺஋తʹٻΊ͍ͯ͘. 16ദ໦խӳ, kv - C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦ http://verifiedby.me/kv/index.html ͜ΕͰ |x| → ∞ ͷ࣌ q-Bessel ؔ਺ΛܭࢉͰ͖ΔΑ͏ʹͳ͚ͬͨͩͰ, ଞͷ q-ಛघؔ਺΍Ұൠ ͷ rϕs ʹ͍ͭͯղܾͰ͖ͯͳ͍. ͨͩ͠ modified q-Bessel ؔ਺ I(2) ν (x; q) ʹ͍ͭͯ͸ Ismail-Zhang (2018) ͷఆཧ 2.1, 1ϕ1 ʹ͍ͭͯ͸ Guindy-Ismail (2016) ͷఆཧ 3.1 Λ࢖͑͹Α͍. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 22 / 54
23. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ (ࢀߟ) ਺஋ੵ෼ʹΑΔํ๏ J(2) ν (x; q) ͸ ℜν > 0 ͷͱ͖࣍ͷੵ෼දࣔΛ࣋ͭ 17 (ଞʹ΋͋Δ 18): 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)∞ . kv ϥΠϒϥϦͰ͸ Bessel ؔ਺Λ਺஋ੵ෼ʹΑΓਫ਼౓อূ෇͖਺஋ܭࢉͯͨ͠ͷ ͱಉ༷ʹ, ਺஋ੵ෼ (Kashiwagi ๏, DE ެࣜ) ʹΑΔ q-Bessel ؔ਺ͷਫ਼౓อূ෇͖ ਺஋ܭࢉ΋ՄೳͰ͋Δ 19. 17Rahman, M. (1987). An integral representation and some transformation properties of q-Bessel functions. Journal of Mathematical Analysis and Applications, 125, 58-71. 18Ismail, M. E., & Zhang, R. (2018). Integral and series representations of q-polynomials and functions: Part I. Analysis and Applications, 16(02), 209-281. 19ۚઘେհ, ؙ໺݈Ұ (2018). Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ๏, Ԡ༻ ྗֶݚڀॴݚڀूձใࠂ 29AO-S7, 1, 49-54. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 23 / 54
24. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ (ࢀߟ) ਺஋ੵ෼ʹΑΔํ๏ ”kv ϥΠϒϥϦ”ʹΑΔਫ਼౓อূ෇͖਺஋ੵ෼ͷྲྀΕ ദ໦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. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 24 / 54
25. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ (ࢀߟ) DE ެࣜ ೋॏࢦ਺ؔ਺ܕੵ෼ެࣜ (DE ެࣜ) ͱ͸, ೋॏࢦ਺ؔ਺ܕͷม਺ม׵ (DE ม׵) ͱ୆ܗެࣜΛ૊Έ߹Θͤͨ਺஋ੵ෼๏Ͱ͋Δ 20,21,22. ͲΜͳ DE ม׵Λࢪ͔͢͸ ੵ෼۠ؒͱඃੵ෼ؔ਺ f ͷ࣋ͭ༏ؔ਺, ͭ·Γ |f(z)| ≤ |F (z)| (∀z ∈ Dd := {z ∈ C : |ℑz| < d < π/2}) ͳΔؔ਺ F (z) ͷछྨʹԠͯ͡࢖͍෼͚͕͞Ε͍ͯΔ. 20Takahashi, H., Mori, M. (1974). Double Exponential Formulas for Numerical Integration. Publications of the Research Institute for Mathematical Sciences, 9(3), 721-741. 21৿ਖ਼෢. (2002). ਺஋ղੳ [ୈ 2 ൛]. ڞཱग़൛. 22ਿݪਖ਼ᰖ, & ࣨాҰ༤. (1994). ਺஋ܭࢉ๏ͷ਺ཧ. ؠ೾ॻళ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 25 / 54
26. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ྵ఺ΛٻΊΔಈػͱઌߦݚڀ ಛघؔ਺ͷྵ఺ʹ͸༷ʑͳԠ༻͕͋Δ. ௚ަଟ߲ࣜͷྵ఺→Gauss ٻੵ (Kreyszig23, Stoer-Bulirsch, Quarteroni-Sacco-Saleri ౳Λࢀর) Bessel ؔ਺ͷྵ఺ → ৽͍͠਺஋ੵ෼ެࣜ (Ogata-Sugihara), ౷ܭͰͷԠ༻ 24 q-ಛघؔ਺ͷྵ఺୳ࡧ͸ q-ಛघؔ਺ͷݚڀʹ໾ཱ͚ͭͩͰͳ͘, ৽ͨͳੵ෼ެࣜ ͷ։ൃʹ΋ͭͳ͕ΔՄೳੑ͕͋Δ. 23Kreyszig, E. (2011). Advanced Engineering Mathematics, 10-th Edition. John Wiley & Sons. 24দຊ༟ߦ, ཱ֬ղੳͱͦͷඍ෼࡞༻ૉͷݚڀ΁ͷԠ༻, Պֶݚڀඅॿ੒ࣄۀ ݚڀ੒Ռใࠂॻ https://kaken.nii.ac.jp/ja/file/KAKENHI-PROJECT-23540183/23540183seika.pdf ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 26 / 54
27. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ q-Bessel ؔ਺ͷྵ఺ ҎԼͷੑ࣭͸ ν > −1 ͷ࣌੒Γཱͭ. Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷྵ఺ ্࣮࣠ʹແݶݸͷྵ఺Λ࣋ͭ (Hahn, 1949) ྵ఺͸͢΂্࣮ͯ࣠ʹ͋Γ, ॏࠜ͸ͳ͍ (Ismail, 1982) Hahn-Exton ͷ q-Bessel ؔ਺ͷྵ఺ Koelink, H., Swarttouw, R. (1994). On the Zeros of the Hahn-Exton q-Bessel Function and Associated q-Lommel Polynomials, Journal of Mathematical Analysis and Applications, 186, 690-710. ্࣮࣠ʹແݶݸͷྵ఺Λ࣋ͭ ྵ఺͸͢΂্࣮ͯ࣠ʹ͋Γ, ॏࠜ͸ͳ͍ ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 27 / 54
28. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ಈػͱઌߦݚڀ ͜Ε·Ͱ༷ʑͳಛघؔ਺ͷྵ఺୳ࡧ͕ߦΘΕ͖͕ͯͨ (Gil-Segura (2014), Segura (2013), · · · ), Մੵ෼ܥ౳ͰݱΕΔ q-ಛघؔ਺ͷྵ఺୳ࡧ͸·ͩͳ͍. q-Newton ๏ͱਫ਼౓อূ෇͖਺஋ܭࢉʹΑΓ q-Bessel ؔ਺ͷྵ఺୳ࡧΛߦ͏ ͱͱ΋ʹ, q-Newton ๏ͷվྑΛఏҊ͢Δ. Bessel ؔ਺ͷྵ఺୳ࡧʹ͍ͭͯ͸, ҎԼͷํ๏͕஌ΒΕ͍ͯΔ. Bessel ؔ਺ʹରͯ͠ Newton ๏Λద༻͢Δ 25 Bessel ؔ਺ͷൺ Jν (x) Jν−1 (x) ʹରͯ͠ Newton ๏Λద༻͢Δ 26 Bessel ؔ਺͸ඍ෼઴ԽࣜΛ࣋ͭͨΊ, Newton ๏ͷద༻͸༰қͰ͋Δ. ͕ͩ q-Bessel ؔ਺ͷಋؔ਺Λܭࢉ͢Δज़͕ͳ͍ͨΊ, Newton ๏͸ద༻Ͱ͖ͳ͍. ͦ͜Ͱࠓճ͸ Newton ๏ͷ୅ΘΓʹ q-Newton ๏Λ࢖͏͜ͱΛߟ͑Δ. 25Garcia, A. (2015). Numerical Methods for Physics, 2nd Edition, Pearson. 26Gil, A., Segura, J., Temme, N. (2007). Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 28 / 54
29. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ q-Newton ๏ q-Newton ๏͸ Newton ๏ͷඍ෼Λ q-ඍ෼ʹஔ͖͔͑ͨ൓෮๏Ͱ͋Δ 27. Definition (q-Newton ๏) xn+1 = xn − f(xn ) Dq f(xn ) , Dq f(x) := f(x) − f(qx) x(1 − q) (q-ඍ෼). q-Newton ๏Ͱ͸ղ͕ಘΒΕΔલʹθϩআࢉ͕ى͖ͯ͠·͏͜ͱ͕͋Δ. q-Newton ๏ʹ͸վྑͷ༨஍͕͋Δ. ͔͜͜Β͸ q-Newton ๏ͷվྑΛߟ͍͕͑ͯ͘, ͦͷલʹ Newton ๏ͷվྑ (ਫ਼౓อূ෇͖਺஋ܭࢉ޲͚) ʹ͍ͭͯݟ͍ͯ͘. 27Rajkovi´ c, P., Stankovi´ c, M., Marinkovi´ c D., (2002). Mean Value Theorems in q-Calculus. Matematiˇ cki Vesnik, 54(3-4), 171-178. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 29 / 54
30. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ ۠ؒ Newton ๏ Newton ๏ͷվྑͱͯ͠, ”۠ؒ Newton ๏”͕஌ΒΕ͍ͯΔ. ۠ؒ Newton ๏ Alefeld, G. (1994). Inclusion Methods for Systems of Nonlinear Equations in: J. Herzberger (Ed.), Topics in Validated Computations, Studies in Computational Mathematics, Elsevier, Amsterdam, 7-26. େੴਐҰ et al. (2018). ਫ਼౓อূ෇͖਺஋ܭࢉͷجૅ, ίϩφࣾ. Newton ๏Ͱղ͕ಘΒΕ͍ͯΔ࣌, ࣍ͷ൓෮: xn+1 = mid(xn) − f(mid(xn)) f′(xn) ʹΑͬͯվྑ͞Εͨղ͕ಘΒΕΔ͜ͱ͕͋Δ. q-Newton ๏Ͱղ͕ಘΒΕ͍ͯΔ࣌, ࣍ͷ൓෮ (q-۠ؒ Newton ๏): xn+1 = mid(xn ) − f(mid(xn )) Dq f(xn ) , Dq f(x) := f(x) − f(qx) x(1 − q) ʹΑͬͯվྑ͞Εͨղ͕ಘΒΕΔ͔΋͠Εͳ͍. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 30 / 54
31. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-۠ؒ Newton ๏ʹΑΓ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷྵ఺୳ࡧΛߦͬͨ. C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦͰ͋Δ”kv ϥΠϒϥϦ”Λ࢖༻ͨ͠. ॳظ஋ x = 20, ν = 1.5, q = 0.7 ͱ࣮ͯ͠ݧΛߦͬͨ. ൓෮ͷ݁Ռ (q-Newton ๏): [4.0077479819329377, 5.674456744153634] ൓෮ͰಘΒΕͨ۠ؒͷԼݶʹΑΔ஋Ҭ: [11.797254862637029, 11.797254862676434] ൓෮ͰಘΒΕͨ۠ؒͷ্ݶʹΑΔ஋Ҭ: [−53.690211419197688, −53.690211418819508] ൓෮ͷ݁Ռ (q-۠ؒ Newton ๏): [4.8965494653086354, 5.1008419887877184] ൓෮ͰಘΒΕͨ۠ؒͷԼݶʹΑΔ஋Ҭ: [6.1224708721105507, 6.1224708722403465] ൓෮ͰಘΒΕͨ۠ؒͷ্ݶʹΑΔ஋Ҭ: [−3.2117521818104465, −3.211752181639532] ൓෮ͰಘΒΕͨ۠ؒͷԼݶͷ஋Ҭ, ্ݶͷ஋Ҭ͸ਖ਼ෛ͕ҟͳΔͨΊ, தؒ஋ͷఆ ཧΑΓ൓෮ͰಘΒΕͨ۠ؒ಺ʹগͳ͘ͱ΋ 1 ͭͷղ͕͋Δ. q-۠ؒ Newton ๏Ͱ վྑ͞ΕͨղΛಘΔ͜ͱ͕Ͱ͖ͨ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 31 / 54
32. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-۠ؒ Newton ๏ʹΑΔղͷݕূఆཧ Theorem (q-۠ؒ Newton ๏ʹΑΔղͷݕূఆཧ) f Λ C1 ڃؔ਺, I Λ༩͑ΒΕͨ۠ؒͱ͠, M ∈ Dq f(I) ͳΔ M ͕θϩͰͳ͍ ͱ͢Δ. ·ͨ, ∃x0 ∈ I ʹରͯ͠, Nq(x0, I) := {x0 − f(x0)/M | M ∈ Dqf(I)} ͱఆΊΔ. Nq (x0 , I) ⊂ I ͳΒ͹ f(x) = 0 ͷղ x∗ ͕Ұҙଘࡏ͢Δ. ͞Βʹ x∗ ∈ Nq (x0 , I) Ͱ͋Δ. ·ͣ q-ඍ෼ੵ෼ͷجຊఆཧΑΓ࣍ͷ౳͕ࣜ੒Γཱͭ: f(x) − f(x0) = ∫ 1 0 Dqf(x0 + t(x − x0))dqt. q-ඍ෼ੵ෼ͷجຊఆཧ Kac, V., Cheung, P. (2001). Quantum Calculus. Springer. f Λݪ఺Ͱ C1 ڃؔ਺ͱ͢Δͱ͕࣍੒Γཱͭ: ∫ b a Dqf(x)dqx = f(b) − f(a). ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 32 / 54
33. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-۠ؒ Newton ๏ʹΑΔղͷݕূఆཧ f(x) − f(x0 ) = M(x)(x − x0 ) (∀x ∈ I), M(x) := ∫ 1 0 Dq f(x0 + t(x − x0 ))dq t ∈ ∫ 1 0 Dq f(I)dq t = Dq f(I). มܗʹ͸ q-chain rule Λ༻͍ͨ. Lemma (q-chain rule) u(x) = αxβ (α, β ∈ C) ͱ͢Δͱ͕࣍੒Γཱͭ: Dq f(u(x)) = (Dqβ f)(u(x)) · Dq (u(x)). Kac, V., Cheung, P. (2001). Quantum Calculus. Springer. ͜͜Ͱؔ਺ g : I → R Λ g(x) := x0 − f(x0 ) M(x) ͱఆΊΔͱ, g ͸࿈ଓͰ͋Δ (∵ M ̸= 0 in I). ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 33 / 54
34. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-۠ؒ Newton ๏ʹΑΔղͷݕূఆཧ ԾఆΑΓ {g(x)|x ∈ I} ⊂ I ͕੒Γཱͭ. Αͬͯ Brouwer ͷෆಈ఺ఆཧΑΓ, g(x∗) = x∗ = x0 − f(x0 ) M(x∗) Λຬͨ͢ෆಈ఺ x∗ ∈ I ͕ଘࡏ͢Δ. Theorem (Brouwer ͷෆಈ఺ఆཧ) compact ತू߹ K ಺ͷ࿈ଓؔ਺͸ K Ͱগͳ͘ͱ΋Ұͭͷෆಈ఺Λ࣋ͭ. Deimling, K. (1985). Nonlinear Functional Analysis, Springer-Verlag. x∗ ͸ f(x∗) = 0 ͷҰҙղͰ͋Δ (Ұҙੑͳ͍ͱ Rajkovi´ c-Stankovi´ c-Marinkovi´ c ͕ࣔͨ͠ q-Rolle ͷఆཧΑΓ۠ؒ I ಺Ͱ M = 0 ʹͳΔ͕͜Ε͸۠ؒ I ಺Ͱ M ̸= 0 ͱ͍͏Ծఆʹ൓͢Δ). ͞Βʹ f(x∗) = 0 ΑΓ, x∗ = x0 − f(x0 ) M(x∗) ∈ Nq (x0 , I) ΋ࣔ͞ΕΔ. 2 ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 34 / 54
35. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ ໰୊఺ɾٙ໰఺ q-Bessel ؔ਺͸ C1 ڃͳͷ͔? q-۠ؒ Newton ๏ʹΑΔղͷݕূఆཧΛ q-Bessel ؔ਺ʹద༻͢Δʹ͸, q-Bessel ؔ਺͕ C1 ڃͰͳ͍ͱ͍͚ͳ͍. Bessel ؔ਺ͷඍ෼઴ԽࣜΑΓ Bessel ؔ਺͕ C1 ڃʹͳΔͷͰ, q-Bessel ؔ਺΋ C1 ڃʹͳΔͱྨਪ͞ΕΔ. ࣮ࡍ, J(3) ν (x; q) ͸ඍ෼ެ͕ࣜಘΒΕ a, C1 ڃʹͳΔ͜ͱ͕ূ໌͞Ε͍ͯΔ. ্͔͠͠ͷΑ͏ʹʮq-ղੳֶͰ͸ͳ͘ (௨ৗͷ) ඍ෼ੵ෼ͷ؍఺͔Β q-ಛघ ؔ਺Λݚڀ͢Δʯ͜ͱʹର͢Δ൷൑΋ࠜڧ͍ b. aKoelink, H. T. Swarttouw, R. F. (1994), On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials, Journal of Mathematical Analysis and Applications, 186: 690-710. bRahman, M. (1989), A note on the orthogonality of Jackson’s q-Bessel function, Canadian Mathematical Bulletin 32, 369-376. q-۠ؒ Newton ๏ͷ໰୊఺ q-۠ؒ Newton ๏Ͱྵ఺͕Ұҙଘࡏ͢ΔൣғΛٻΊΒΕΔ͜ͱ͕෼͔ͬͨ. ͔۠ؒ͠͠෯ͷ૿େ͕཈͑ΒΕ͍ͯͳ͍. ۠ؒ෯ͷ૿େΛ཈͑ͳ͕Βྵ఺ΛٻΊΔํ๏͸ͳ͍ͩΖ͏͔? ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 35 / 54
36. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ Krawczyk ๏ ؆қ Newton ๏ʹฏۉ஋ܗࣜΛద༻ͯ͠ಘΒΕΔ Krawczyk ๏ʹண໨͢Δ 28, 29. ؆қ Newton ๏ (ʮେੴਐҰ. (1997). ඇઢܗղੳೖ໳, ίϩφࣾ.ʯ͕ৄ͍͠) xn+1 = xn − f(xn)/f′(x0) ฏۉ஋ܗࣜ େੴਐҰ. (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. c := mid(I) ͱ͢Δ. f(I) Λ௚઀ධՁ͢Δ୅ΘΓʹ f(c) + f′(I)(I − c) Λܭࢉ͢Δ. Krawczyk ๏ େੴਐҰ et al. (2018). ਫ਼౓อূ෇͖਺஋ܭࢉͷجૅ, ίϩφࣾ. xn+1 = mid(xn) − f(mid(xn)) f′(x0) + ( 1 − f′(xn) f′(x0) ) (xn − mid(xn)). 28Krawczyk, R. (1969). Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehler-schranken, Computing 4, 187-201. 29Krawczyk, R. (1969). Fehlerabsch¨ atzung reeller Eigenwerte und Eigenvektoren von Matrizen, Computing 4, 281-293. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 36 / 54
37. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ ฏۉ஋ܗࣜͷܭࢉྫ f(x) := x − x ͱ͢Δ. ͜ͷͱ͖۠ؒԋࢉͰ͸ f([a, b]) ⊂ [a, b] − [a, b] = [a − b, b − a] (a < b) ͱͳͬͯ͠·͏. ҰํͰฏۉ஋ܗࣜͰ͸ f′(x) = 0 ͳͷͰ F ([a, b]) = 0 + 0 ∗ [a − c, b − c] = 0 ͱͳΔ. ۠ؒԋࢉ͚ͩͰ͸ܾͯ͠ॖখ͢Δ͜ͱͷͳ͔ͬͨ۠ؒ෯͕͜ͷ৔߹ʹ͸ ฏۉ஋ܗࣜΛ༻͍Δ͜ͱͰॖখ͢Δ͜ͱ͕෼͔ͬͨ. Ұൠʹ f′([a, b]) ͕খ͘͞ [a, b] ͕খ͞ͳ෯ͷͱ͖ฏۉ஋ܗࣜ͸Α͍෯Λ༩͑Δ. େੴਐҰ (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 37 / 54
38. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ Krawczyk ๏ Krawczyk ๏ͷಛ௃ ॳظ஋Λे෼ۙ͘ʹͱΒͳ͍ͱऩଋ͠ͳ͍ (؆қ Newton ๏͕ݩ͔ͩΒ). ͔۠ؒ͠͠෯ͷ૿େ͸཈͑ΒΕΔ (ฏۉ஋ܗࣜͷޮՌ) ͜͜Ͱ͸ Krawczyk ๏ͷ୅ΘΓʹ࣍ͷ൓෮Λߦ͏. xn+1 =mid(xn ) − f(mid(xn )) Dq f(x0 ) + ( 1 − Dq f(xn ) Dq f(x0 ) ) (xn − mid(xn )). ͜ͷ൓෮Λ q-Krawczyk ๏ͱݺͿ͜ͱʹ͢Δ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 38 / 54
39. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-Krawczyk ๏ʹΑΔ਺஋࣮ݧ q-Krawczyk ๏ʹΑΓ Jackson ͷୈ 2 छ q-Bessel ؔ਺ͷྵ఺୳ࡧΛߦͬͨ. C++ ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦͰ͋Δ”kv ϥΠϒϥϦ”Λ࢖͍ͬͯΔ. ॳظ஋ x = 1, ν = 1.5, q = 0.8 ൓෮ͷ݁Ռ (q-Krawczyk ๏, ൓෮ 20 ճ): [0.97640148781825686, 0.97640148782929049] ൓෮ͰಘΒΕͨ۠ؒͷԼݶʹΑΔ஋Ҭ: [2.8695851995687983 × 10−12, 3.1844884133952458 × 10−11] ൓෮ͰಘΒΕͨ۠ؒͷ্ݶʹΑΔ஋Ҭ: [−3.1852420577536858 × 10−11, −2.7380333548111838 × 10−12] ൓෮ͰಘΒΕͨ۠ؒͷԼݶͷ஋Ҭ, ্ݶͷ஋Ҭ͸ਖ਼ෛ͕ҟͳΔͨΊதؒ஋ͷఆཧ ΑΓ൓෮ͰಘΒΕͨ۠ؒ಺ʹগͳ͘ͱ΋ 1 ͭͷղ͕͋Δ. q-Krawczyk ๏Ͱྵ఺ ଘࡏൣғ͕ಘΒΕ্ͨʹ, ۠ؒ෯ͷ૿େ΋཈͑ΒΕ͍ͯΔ. q-ྨࣅͯ͠΋ฏۉ஋ ܗࣜͷޮՌ͕ग़͍ͯΔͷͩΖ͏. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 39 / 54
40. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ ໰୊఺ q-۠ؒ Newton ๏, q-Krawczyk ๏Ͱྵ఺ଘࡏൣғΛܭࢉՄೳʹͳͬͨ 30. ͔͠͠൓෮ 1 ճʹ͖ͭ q-Bessel ؔ਺Λ 3 ճҎ্ܭࢉ͠ͳ͍ͱ͍͚ͳ͍ͷ͕ ऑ఺Ͱ͋Δ. ൓෮ 1 ճʹ͖ͭ q-Bessel ؔ਺Λ 2 ճܭࢉ͢Δ͚ͩͰࡁ·ͤΒΕͳ͍͔? 30ͪͳΈʹ q-Newton Kantorovich ͷఆཧ͸Ҏલ͔Β͋Δ. Rajkovi´ c, P. M., Marinkovi´ c, S. D., & Stankovi´ c, M. S. (2005). On q-Newton-Kantorovich method for solving systems of equations. Applied Mathematics and Computation, 168(2), 1432-1448. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 40 / 54
41. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-Bessel ؔ਺ͷ࣮ࠜ୳ࡧ xn+1 = xn − arctan ( Jν (xn ) Jν−1 (xn ) ) ͸ॳظ஋Λे෼ۙ͘ͱΔͱ Bessel ؔ਺ͷྵ఺ʹऩଋ͢Δͱ͍͏͜ͱ͕஌ΒΕͯ ͍Δ (Gil-Segura-Temme, 2007). ͜ͷํ๏ͳΒ͹ 1 ൓෮Ͱ Bessel ؔ਺Λ 2 ճܭࢉ ͢Δ͚ͩͰࡁ·ͤΒΕΔ. ༧૝ xn+1 = xn − arctan ( Jν (xn ; q) Jν−1 (xn ; q) ) ͸ॳظ஋Λे෼ۙ͘ͱΔͱ q-Bessel ؔ਺ͷྵ఺ʹऩଋ͢ΔͷͰ͸? ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 41 / 54
42. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ ࣮ݧ݁Ռ Hahn-Exton ͷ q-Bessel ؔ਺, ॳظ஋ x = 6.5, ν = 4.5, q = 0.8 ൓෮ͷ݁Ռ (൓෮ 5 ճ): [0.09581004804906934, 2.2425868252899446] ൓෮ͰಘΒΕͨ۠ؒͷԼݶʹΑΔ஋Ҭ: [0.0014753173015071661, 0.0014753173015524919] ൓෮ͰಘΒΕͨ۠ؒͷ্ݶʹΑΔ஋Ҭ: [−4.5808580979457512, −4.580858022855625] ൓෮ͰಘΒΕͨ۠ؒͷԼݶͷ஋Ҭ, ্ݶͷ஋Ҭ͸ਖ਼ෛ͕ҟͳΔͨΊதؒ஋ͷఆཧ ΑΓ൓෮ͰಘΒΕͨ۠ؒ಺ʹগͳ͘ͱ΋ 1 ͭղ͕͋Δ. ྵ఺ଘࡏൣғ͕ಘΒΕͨ. xn+1 = xn − arctan ( Jν (xn ; q) Jν−1 (xn ; q) ) ͸ॳظ஋Λे෼ۙ͘ͱΔͱ q-Bessel ؔ਺ͷྵ఺ʹऩଋ͢Δ͔΋͠Εͳ͍ͱ͍͏ ࣔࠦΛಘͨ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 42 / 54
43. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ xn+1 = xn − arctan ( Jν (xn ; q) Jν−1 (xn ; q) ) ͸ॳظ஋Λे෼ۙ͘ͱΔͱ q-Bessel ؔ਺ͷྵ఺ʹऩଋ͢Δ͔΋͠Εͳ͍ͱ͍͏ ࣔࠦΛಘͨ. ΋͔ͨ͠͠Β, xn+1 = xn − arctanq ( Jν (xn ; q) Jν−1 (xn ; q) ) ΋ॳظ஋Λे෼ۙ͘ͱΔͱ q-Bessel ؔ਺ͷྵ఺ʹऩଋ͢ΔͷͰ͸? ͜Ε·Ͱ༷ʑͳ q-ಛघؔ਺ (q-Bessel, q-exp, q-sin,· · · ) ͕ݚڀ͞Ε͖͕ͯͨ, q-arctan ʹؔ͢Δݚڀ͸ͳ͍. →q-arctan ΛͲ͏΍ͬͯఆٛ͢Δ? ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 43 / 54
44. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-arctan ͷఆٛ (ͦͷ 1) Definition (Gauss ͷ௒زԿؔ਺ 2 F1 , Pochhammer ه߸ (a)n ) 2F1 (a, b; c; x) := ∞ ∑ n=0 (a)n(b)nxn (c)nn! , |x| < 1, (a)n := a(a + 1) · · · (a + n − 1). ݪԬتॏ. (2002). ௒زԿؔ਺. ே૔ॻళ. arctan(x) = x 2F1 ( 1, 1 2 ; 3 2 ; −x2 ) , (|x| < 1) (Taylor ల։), arctan(x) = sgn(x) π 2 − arctan ( 1 x ) , arctan(±1) = ± π 4 ΑΓ arctanq(x) :=        x 2ϕ1 ( q, q 1 2 ; q 3 2 ; −x2 ) , (|x| < 1), ± πq 4 (x = ±1), sgn(x)πq 2 − 1 x 2ϕ1 ( q, q 1 2 ; q 3 2 ; − 1 x2 ) ͱఆΊΔ (πq := ( Γq ( 1 2 ))2 , Γq (x) := (q;q)∞ (qx;q)∞ (1 − q)1−x). Γq (x) ͸ (Jackson ͷ) q-gamma ؔ਺Ͱ͋Δ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 44 / 54
45. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-arctan ͷఆٛ (ͦͷ 2) Theorem (arctan ͷผදݱ) arctan(x) = x 1 + x2 2 F1 ( 1, 1; 3 2 ; x2 1 + x2 ) , x ∈ R. Castellanos, D. (1988), The Ubiquitous Pi. Part I. Math. Mag. 61, 67-98. ্ͷఆཧ͔Β, q-arctan Λ arctanq (x) := x 1 + x2 2 ϕ1 ( q, q; q 3 2 ; q; x2 1 + x2 ) , x ∈ R. ͱఆΊΔ͜ͱ΋Ͱ͖Δ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 45 / 54
46. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ q-arctan ͷఆٛ (q-ੵ෼ܕ) arctan(x) = ∫ x 0 1 1 + t2 dt ͳͷͰ, arctanint q (x) := ∫ x 0 1 1 + t2 dq t = x(1 − q) ∞ ∑ n=0 qn 1 + x2q2n ͱఆٛ͢Δ͜ͱ΋ՄೳͰ͋Δ. ∑ ∞ n=0 qn 1+x2q2n ͷଧ੾ΓޡࠩΛධՁ͢Δ. T (n) := qn 1+x2q2n ͱ͓͘ͱ x ∈ R, n ≥ N ͷͱ͖, T (n + 1) T (n) = q 1 + x2q2n 1 + x2q2n+2 ≤ q(1 + x2q2N ) =: D Ͱ͋Γ, D < 1 ͷͱ͖, ∞ ∑ n=N T (n) ≤ T (N) 1 − D ͱͳΔ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 46 / 54
47. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ ਺஋࣮ݧ ͜͜·Ͱಋೖͨ͠ 3 ͭͷ q-arctan ʹ͍ͭͯ਺஋࣮ݧΛߦͬͨ. x = 1.5, q = 0.1 1 ൪໨:[25.425717946994236, 25.425717946998969] 2 ൪໨:[1.42175327653972, 1.42175327653992] ੵ෼ܕ:[0.56241091538066489, 0.56241091538067634] 3 ͭͷ q-arctan ͸Ұൠʹ͸ҟͳΔ஋ΛऔΔؔ਺ͨͪͰ͋Δ͜ͱ͕෼͔ͬͨ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 47 / 54
48. ݚڀ੒Ռ (म࿦ͷओ݁Ռ) q-Bessel ؔ਺ͷྵ఺୳ࡧ ࣮ݧ݁Ռ q-arctanʢੵ෼ܕʣΛ࢖࣮ͬͯݧΛߦͬͨ. Jackson ͷ 2 छ q-Bessel ؔ਺, ॳظ஋ x = 3.3, ν = 1.5, q = 0.5 ൓෮ͷ݁Ռ (൓෮ 11 ճ): [3.3617264194240701, 3.361727002402478] ൓෮ͰಘΒΕͨ۠ؒͷԼݶʹΑΔ஋Ҭ: [2.7636118073791531 × 10−7, 2.7636124059863484 × 10−7] ൓෮ͰಘΒΕͨ۠ؒͷ্ݶʹΑΔ஋Ҭ: [−1.1506353552420116 × 10−6, −1.1506352954317914 × 10−6] Hahn-Exton ͷ q-Bessel ؔ਺, ॳظ஋ x = 6.5, ν = 4.5, q = 0.8 ൓෮ͷ݁Ռ (൓෮ 4 ճ): [0.52973884045367647, 2.2374167684626767] ൓෮ͰಘΒΕͨ۠ؒͷԼݶʹΑΔ஋Ҭ: [0.4980629706212637, 0.49806297107742914] ൓෮ͰಘΒΕͨ۠ؒͷ্ݶʹΑΔ஋Ҭ: [−4.1153699782227538, −4.1153699110736381] ൓෮ͰಘΒΕͨ۠ؒͷԼݶͷ஋Ҭ, ্ݶͷ஋Ҭ͸ਖ਼ෛ͕ҟͳΔͨΊதؒ஋ͷఆཧ ΑΓ൓෮ͰಘΒΕͨ۠ؒ಺ʹগͳ͘ͱ΋ 1 ͭղ͕͋Δ. ྵ఺ଘࡏൣғ͕ಘΒΕͨ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 48 / 54
49. ·ͱΊͱ՝୊ ͜͜·Ͱͷ·ͱΊͱ՝୊ ͜͜·Ͱͷ·ͱΊ q-Bessel ؔ਺ͱͦͷྵ఺ͷਫ਼౓อূ๏Λཱ֬ͨ͠ (|ν| → ∞ ͷ࣌Λআ͘). q-Bessel ؔ਺ͷྵ఺୳ࡧ q-۠ؒ Newton ๏Ͱྵ఺͕Ұҙଘࡏ͢ΔൣғΛٻΊΔ. ↓ q-Krawczyk ๏Ͱྵ఺͕ଘࡏ͢ΔൣғΛڱΊΔ. ࠓޙͷ՝୊ ॳظ஋ͷͱΓํΛ޻෉Ͱ͖ͳ͍͔ ? C ্ͷྵ఺Λݟ͚͍ͭͨ. xn+1 = xn − arctan ( Jν (xn;q) Jν−1 (xn;q) ) ͷऩଋΛࣔ͢. ࠓޙ΋ಛघؔ਺, ਺஋ੵ෼ͷਫ਼౓อূʹؔ࿈͢ΔݚڀΛ͍ͯ͜͠͏ͱߟ͍͑ͯΔ. ·ͨՄੵ෼ΞϧΰϦζϜͷݚڀʹ΋ڵຯɾؔ৺Λ͍࣋ͬͯΔ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 49 / 54
50. ଞʹम࿦౳Ͱ΍ͬͨ͜ͱ (ࢀߟ) ଞʹम࿦౳Ͱ΍ͬͨ͜ͱ q-gamma ؔ਺ͷ਺஋ੵ෼ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉ →2017 ೥ 12 ݄ͷֶձൃද, Ismail (1981) ͕ಋग़ͨ͠ੵ෼දࣔΛ࢖༻ͨ͠. Jackson ͷୈ 2 छ q-Bessel ؔ਺ͱ Hahn-Exton ͷ q-Bessel ؔ਺ͷແݶ۠ؒ ੵ෼දࣔ (Ismail-Zhang, 2018) ʹର͠ DE ม׵ exp(t − exp(−t))31 Λద༻ ͨ͠ࡍͷܭࢉਫ਼౓ʹؔ͢Δݚڀ →M1 Ͱͷதؒൃදͷ಺༰. Մੵ෼ܥͰݱΕΔପԁ௒زԿؔ਺ 32,33 (Gasper-Rahman, Spiridonov Λࢀর) ͷਫ਼౓อূ෇͖਺஋ܭࢉ →2017 ೥ߴڮݚ&ؙ໺ݚ߹ಉθϛͰͷൃද಺༰ΛՃචͨ͠. ߦྻ q-ಛघؔ਺ 34,35 ͷਫ਼౓อূ෇͖਺஋ܭࢉ →Higham, Functions of Matrices: Theory and Computation ʹ৮ൃ͞Εͨ, M2 Ͱͷݚڀ. 31Tanaka, K. I., Sugihara, M., Murota, K., & Mori, M. (2009). Function classes for double exponential integration formulas. Numerische Mathematik, 111(4), 631-655. 32Date-Jimbo-Kuniba-Miwa-Okado (1987), Frenkel-Turaev (1997) ʹΑͬͯॳΊͯಋೖ͞Εͨ. 33ପԁಛघؔ਺ͷྫͱͯ͠ Ruijsenaars (1997) ͕ಋೖͨ͠ପԁ gamma ؔ਺΋ڍ͛ΒΕΔ. 34Salem, A. (2012). On a q-gamma and a q-beta matrix functions. Linear and Multilinear Algebra, 60(6), 683-696. 35Salem, A. (2014). The basic Gauss hypergeometric matrix function and its matrix q-difference equation. Linear and Multilinear Algebra, 62(3), 347-361. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 50 / 54
51. ଞʹम࿦౳Ͱ΍ͬͨ͜ͱ ߦྻ q-ಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ ߦྻ஋ؔ਺ ਺஋ղੳ (ྫ͑͹ exponential integrator), ౷ܭֶ౳ͰԠ༻͕ ͋Δ. q-ಛघؔ਺ Մੵ෼ܥ౳ͷ਺ཧ෺ཧͰॏཁࢹ͞ΕΔ. ↘ ↙ ߦྻ q-ಛघؔ਺͸ݚڀ͞ΕΔ΂͖ॏཁͳؔ਺͔΋͠Εͳ͍. (A; q)n := n−1 ∏ k=0 (I−Aqk), (A; q)∞ := lim n→∞ (A; q)n (ߦྻ q-Pochhammer ه߸) Γq (A) := (q; q)∞ (qA; q)−1 ∞ (1 − q)I−A, |q| < 1 (ߦྻ q-gamma ؔ਺) ↓ ߦྻ q-ಛघؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉ͕ॏཁʹͳΓ͏Δ. ɾߦྻ஋ؔ਺ͷཧ࿦ʹ͍ͭͯ͸ʮߦྻͷؔ਺ͱδϣϧμϯඪ४ܗʯ (ઍ༿ࠀ༟, 2010) ΋ৄ͍͠. ɾexponential integrator ͸࿈ཱ ODE ͷ਺஋ղ๏Ͱ͋Δ. ɾexp(A), log(A) ͷਫ਼౓อূ෇͖਺஋ܭࢉ͸ Miyajima (2019) ʹΑͬͯݚڀ͞Ε͍ͯΔ. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 51 / 54
52. ࢀߟจݙ ࢀߟจݙ q-ಛघؔ਺ΛؚΉಛघؔ਺શൠʹৄ͍͠ DLMF: Digital Library of Mathematical Functions (ถࠃ NIST ͕࡞੒) Andrews, G. E., Askey, R., & Roy, R. (2000). Special Functions (Encyclopedia of Mathematics and its Applications Vol. 71). Cambridge University Press. q-ಛघؔ਺ʹৄ͍͠ॻ੶ Gasper, G. & Rahman, M. (2004). Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications Vol. 96). Cambridge University Press.→11 ষͰ͸ପԁ௒زԿؔ਺ʹ͍ͭͯ΋ղઆ͞Ε͍ͯΔ. Andrews, G. E. (1986). q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra (No. 66). American Mathematical Society. Exton, H. (1983). q-Hypergeometric Functions and Applications. Horwood. q-ಛघؔ਺ʹؔ͢Δ Review هࣄ Koelink, E. (2018). q-Special Functions, Basic Hypergeometric Series and Operators. arXiv preprint arXiv:1808.03441. Koornwinder, T. H. (2005). q-Special Functions, an Overview. arXiv preprint math/0511148. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 52 / 54
53. ࢀߟจݙ ࢀߟจݙ q-ղੳֶʹৄ͍͠ॻ੶ Ernst, T. (2012). A Comprehensive Treatment of q-calculus. Springer Science & Business Media. Ernst, T. (2000). The History of q-Calculus and a New Method. Department of Mathematics, Uppsala University. Kac, V., & Cheung, P. (2001). Quantum Calculus. Springer Science & Business Media.→ h-ղੳֶʹ͍ͭͯ΋ղઆ͞Ε͍ͯΔ. Hahn-Exton ͷ q-Bessel ؔ਺ʹৄ͍͠ Swarttouw, R. F. (1992), The Hahn-Exton q-Bessel Function, PhD Thesis, Delft Technical University. (q-) ௚ަଟ߲ࣜʹৄ͍͠ॻ੶ Ismail, M. E. (2006). Classical and Quantum Orthogonal Polynomials in One Variable. (Encyclopedia of Mathematics and its Applications) Cambridge University Press. Koekoek, R., Lesky, P. A., & Swarttouw, R. F. (2010). Hypergeometric Orthogonal Polynomials and their q-Analogues. Springer Science & Business Media. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 53 / 54
54. ࢀߟจݙ ಛघؔ਺Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖ΔϥΠϒϥϦ INTLAB (Interval Laboratory)36→gamma ؔ਺ͱޡࠩؔ਺͕ܭࢉͰ͖Δ. kv ϥΠϒϥϦ →gamma ؔ਺ͱ Bessel ؔ਺, Airy ؔ਺౳͕ܭࢉͰ͖Δ. Arb - a C library for arbitrary-precision ball arithmetic37 →Fredrik Johansson ʹΑͬͯ։ൃ͞ΕͨϥΠϒϥϦͰ͋Γ, ༷ʑͳಛघؔ਺ (ପԁؔ਺, ௒زԿؔ਺౳) Λਫ਼౓อূ෇͖਺஋ܭࢉͰ͖Δ. 36S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999. 37Johansson, F. (2016). Computing Hypergeometric Functions Rigorously. arXiv preprint arXiv:1606.06977. Arb ͷ࣮૷͸͜ͷ࿦จ౳ʹج͍͍ͮͯΔ. ͜ͷ࿦จ͸௒زԿؔ਺ pFq ͷଧ੾ΓޡࠩΛධՁ͢Δख๏ Λܝࡌ͓ͯ͠Γ, q-௒زԿؔ਺ rϕs ͷଧ੾ΓޡࠩΛධՁ͢Δࡍʹࢀরͨ͠. ۚઘେհ, ؙ໺݈Ұ (ૣҴాେֶ) q-Bessel ؔ਺ͷਫ਼౓อূ෇͖਺஋ܭࢉɾྵ఺୳ࡧ 2019 ೥ 5 ݄ 24 ೔ 54 / 54