Σk=1. d n () z = () z Σ dz n. d n. 1 = () z. z (0.1 - ) d f 2 = dz 0 () z = 1 () z, log. f n = n-2 () z = n-1 () z. e log z. = e.
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1 Series Expasio of Gamma Fuctio & the Reciprocal. Taylor Expasio aroud a Higher Derivative of Gamma Fuctio The formula of the higher derivative of the gamma fuctio & the reciprocal was discovered by Masayuki Ui i December 06. ( See Higher Derivative of Compositio Sec.3 ) I reproduce it here as follows. Formula..0 ( Masayuki Ui ) Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressios hold. d () () d k d d () () B,k 0 (), (),, - () (0. + ) k (-) k B,k 0 (), (),, - () (0. - ) Whe f() log(), d f log() d 0 (), d f d 0 () (), d f - () - () d Substitutig these ad g k e f k,,3, for the followig Faà di Bruo's Formula o a composite fuctio we obtai gf( x) gk B r,kf, f,, f e log e log B,k 0 (), (),, - () (0. + ) k Whe f() -log(), i a similar way, we obtai e -log e -log (-) k B,k 0 (), (),, - () k (0. - ) Usig this formula, we ca perform the Taylor expasio of the gamma fuctio ad the reciprocal /. Where, we ca ot perform the Taylor expasio aroud a 0,-,-,-3,. Because, at these poits, the differetial coefficiets are or 0. Formula.. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressio holds for a s.t. a 0,-,-,-3,. () ( a ) + c () a ( -a)! (.) - -
2 c () a ( a) k B,k 0 () a, () a,, - () a,,3, ca be expaded to Taylor series aroud a 0,-,-,-3, as follows. () ( a ) + () a ( -a)! Applyig Formula..0 (0. + ) to this ad replacig () a with c a, we obtai the desired formula. Example: Taylor expasio aroud ( symbolic calculatio ) Accordig the formula, is expaded to Taylor series aroud. The polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. The expasio util the 3rd term is as follows. O the other had, whe is expaded to series aroud usig the fuctio Series[] of Mathematica it is as follows. Though they seem to be differet, they are the same thig. Ideed, if 0 []-, [] /6- are substituted for f t,,3, it is as follows. - -
3 Formula.. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressio holds for a s.t. a 0,-,-,-3,. () + () a c () a () a k c () a ( -a)! (-) k B,k 0 () a, () a,, - () a / ca be expaded to Taylor series aroud a 0,-,-,-3, as follows. () + () a () ( -a) a! Applyig Formula..0 (0. - ) to this ad replacig the differetial coefficiet with c a, we obtai the desired formula. Example: Taylor expasio aroud ( umeric calculatio ) Accordig the formula, / is expaded to Taylor series aroud. The polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. If the right side is expaded util 0 terms ad is illustrated with the left side, it is as follows. Both sides are exactly overlapped ad the left side (blue) is ivisible. (.) - 3 -
4 . Lauret Expasio of Gamma Fuctio & the Reciprocal We ca ot perform the Maclauri expasio of the gamma fuctio ad the reciprocal /. But, we ca perform the Maclauri expasio of the + ad the reciprocal /+. Usig this, we ca perform the Lauret expasio of the ad the reciprocal / aroud 0. Formula.. ( Lauret expasio ) Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressio holds () + c k c -! B,k 0 (), (),, - (),,3, (.) + ca be expaded to Maclauri series as follows. ( + ) + () k ()! B,k 0 (), (),, - () Replacig () () with c ad dividig both sides by, ( +) c k + c -! B,k 0 (), (),, - () Sice +, ( ), we obtai the desired expressio. Numeric Calculatio Accordig the formula, is expaded to Lauret series aroud 0. The polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. The expasio util the 4th term is as follows
5 O the other had, is expaded to series aroud 0 usig the fuctio Series[] of Mathematica as follows. This is cosistet with the above exactly. I additio, if f t is expaded util 0 terms ad is illustrated with, it is as follows. Both are almost overlapped Formula.. ( reciprocal Lauret expasio ) Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressio holds () + c +! c ( -) k B k,k 0 (), (),, - (),,3, (.) + ca be expaded to Maclauri series as follows. ( +) ( +) + 0 () ( +) k 0! (-) k B,k 0 (), (),, - () Replacig the differetial coefficiet with c ad multiplyig both sides by, ( +) + + c! - 5 -
6 k c () (-) k B,k 0 (), (),, - () Sice +, ( ), we obtai the desired expressio. Symbolic Calculatio Accordig the formula, / is expaded to series aroud 0. The polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. The expasio util the 4th term is as follows. O the other had, whe / is expaded to series aroud 0 usig the fuctio Series[] of Mathematica, it is as follows. Though they seem to be differet, they are the same thig. Ideed, if 0 [] -, [] /6, 3 [] 4 /5 are substituted for f t,4, it is as follows
7 .3 Maclauri Expasio Formula.3. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressios hold ( +) ( -) ( +/) ( -/) + c! + (-) +! c! (3. - ) (3. + ) c (3. - ) + (-) c (3. + )! c ( -) k B k,k 0 (), (),, - (),,3, /+ ca be expaded to Maclauri series as follows. + ( +) ( +) 0! ( +) 0 k () (-) k B,k 0 (), (),, - () Replacig the differetial coefficiet with c we obtai (3. - ). Ad reversig the sig we obtai (3. + ). Replacig with / i (3. - ) we obtai (3. - ). Ad reversig the sig we obtai (3. + ). Example: /( +/ )( symbolic calculatio ) Accordig the formula, /+/ is expaded to series aroud 0. The polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. The expasio util the 4th term is as follows
8 O the other had, whe f is expaded to series aroud 0 usig the fuctio Series[] of Mathematica, it is as follows. Though they seem to be differet, they are the same thig. Ideed, if 0 [] -, [] /6, 3 [] 4 /5 are substituted for f m,4, it is as follows. Formula.3. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressios hold ( + )/ ( - )/ c k + c (3.3 - )! + (-) c (3.3 + )! - k B,k 0,,,- /( + )/ ca be expaded to Maclauri series as follows. ( + )/ 0 ( + )/ 0! The first term is ( + )/ () 0 0 ( +0 )/ The secod ad the subsequet terms are obtaied usig Formula..0 d d () () k,,3, (-) k B,k 0 (), (),, - () - 8 -
9 as follows. i.e. ( + )/ ( + )/ ( + )/ + / () 0 () 0 0 k k k + k - k B,k k B,k 0 +0, +0 - k B,k 0,,,-,,3, - k B,k 0,,,- Multiplyig by both sides ad replacig the ier with c, we obtai (3.3 - ). I a similar way, (3.3 + ) is also obtaied. ( See the proof of Formula.3.3 )! Example: /( - )/ ( umeric calculatio ) Accordig the formula, /- / is expaded to Maclauri series. The Bell polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. If the right side is expaded util 5 terms ad is illustrated with the left side, it is as follows. Both sides are exactly overlapped ad the left side (blue) is ivisible almost
10 Formula.3.3 Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressio holds + ( 3- )/ (-) c (3.4 + )! c k - k B,k ,,,-,,3, /( 3- )/ ca be expaded to Maclauri series as follows. ( 3- )/ 0 ( 3- )/ 0! The first term is ( 3- )/ () 0 0 ( 3-0 )/ The secod ad the subsequet terms are obtaied usig Formula..0 d d as follows. i.e. () ( 3- )/ ( 3- )/ ( 3- )/ 3- / () k () 0 () 0 0 (-) k B,k 0 (), (),, - () k k k + k - k B,k k B,k 0 3-0, k B,k ,,,- - k B,k ,,,- Multiplyig by / both sides ad replacig the ier with c, we obtai (3.4 + ). (-),,3, (-)! Symbolic Calculatio Accordig the formula, /( 3- )/ is expaded to Maclauri series. The Bell polyomial B,k f, f,is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. The expasio util the 4th term is as follows
11 O the other had, whe f is expaded to series aroud 0 usig the fuctio Series[] of Mathematica, it is as follows. Though they seem to be differet, they are the same thig. Ideed, if [ 3/ ] /-4 ad 3 [ 3/ ] 4-96 are substituted for f m,4, it is as follows. - -
12 .4 Taylor Expasio aroud (Part ) Formula.4. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressios hold c + () ( -)! (4. - ) + (-) c ( -) ( -)! (4. + ) c + ( + )/ ( -)! (4. - ) + ( 3- )/ (-) c ( -)! (4. + ) -( - ) - (-) c ( -) + ( -)! (4.5 + ) c ( -) k B k,k 0 (), (),, - (),,3, Replacig with - i Formula.3. (3. - ) ~ (3. + ), we obtai (4. - ) ~ (4. + ). Multiplyig both sides of (4. + ) by -, we obtai (4.5 + ). Strictly, (4.5 + ) should be called reciprocal Lauret expasio. ( + )/ ( umeric calculatio ) Example: / Accordig the formula, /+ / is expaded to Taylor series aroud. The Bell polyomial f, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. B,k If the right side is expaded util 5 terms ad is illustrated with the left side, it is as follows. Both sides are exactly overlapped ad the left side (blue) is ivisible almost. - -
13 Formula.4. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressio holds ( +) + c ( -)! c ( -) k B k,k 0 (), (),, - (),,3, (4.5 - ) /+ ca be expaded to Taylor series as follows. ( +) ( +) + () ( +) k ( -)! Replacig the differetial coefficiet with c we obtai (4.5 - ). Symbolic Calculatio (-) k B,k 0 (), (),, - () Accordig the formula, / + is expaded to Taylor series. The polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. The expasio util the 3rd term is as follows. O the other had, whe /+ is expaded to series aroud usig the fuctio Series[] of Mathematica, it is as follows
14 Though they seem to be differet, they are the same thig. Ideed, if 0 [] -, [] /6- are substituted for f t,3, it is as follows
15 .5 Taylor Expasio aroud (Part ) Formula.5. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressios hold ( /) ( -/) c k + + (-) c ( -) (5.3 - )! c ( -) (5.3 + )! - k B,k 0,,,-,,3, / ca be expaded to Taylor series as follows. / ( /) i.e. ( /) ( /) / 0 () 0 Multiplyig both sides by we obtai (5.3 + ). ( /) ( /) ( -)! k + k - k B,k 0,,,- - k B,k 0,,,-,,3, -! ad replacig the ier with c, we obtai (5.3 + ). I a similar way, Example: /( -/ )( umeric calculatio ) Accordig the formula, /-/ is expaded to Taylor series aroud. The Bell polyomial f, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. B,k If the right side is expaded util 5 terms ad is illustrated with the left side, it is as follows. Both sides are exactly overlapped ad the left side (blue) is ivisible almost
16 Formula.5. Whe is the gamma fuctio, is the polygamma fuctio ad B,kf, f, are Bell polyomials, the followig expressio holds ( +/) c k + c ( -) (5.6 - )! - k B,k ,,,-,,3, /+ / ca be expaded to Taylor series as follows. i.e. ( +/) ( +/) ( +/) +/ 0 () 0 ( +/) ( 3/) ( -)! k + k 3 - k B,k 0,,, - 3 3,,3, - k B,k ,,,- Multiplyig both sides by / ad replacig the ier with c, we obtai (5.6 - ). -! Symbolic Calculatio Accordig the formula, /+/ is expaded to Taylor series. The polyomial B,kf, f, is geerated usig the fuctio BellY[] of formula maipulatio software Mathematica. The expasio util the 3rd term is as follows
17 O the other had, whe f is expaded to series aroud usig the fuctio Series[] of Mathematica, it is as follows. Though they seem to be differet, they are the same thig. Ideed, if [3/] /-4 are substituted for f t,3, it is as follows Added Formula.3. & Formula.3.3. Alie's Mathematics Kao. Koo - 7 -
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