Extension of Summation Formulas involving Stirling series

Transcription

1 Etension of Summation Formulas involving Stirling series Raphael Schumacher arxiv: v [mathnt] 6 May 06 Abstract This paper presents a family of rapidly convergent summation formulas for various finite sums of the form f(), where is a positive real number Introduction In this paper we will use the Euler-Maclaurin summation formula [3, 5] to obtain rapidly convergent series epansions for finite sums involving Stirling series [] Our ey tool will be the so called Weniger transformation [] For eample, one of our summation formulas for the sum, where R is ( ) 3 ( ) l= ( )l (l 3)!! l (l)! S() (l)b l({}) = 3 3 4π ζ = 3 3 4π ζ B ({}) ()() () ) ( ) ( {} 4 {} {} 4 4) () ( 3 ( 4 {}3 3 6 {} {} 48 4) ()() ( 8 {} {} {}3 9 3 {} ()()(3)(4) ( 64 {}4 3 3 {}3 64 {} 7 53 {} 6 640) ()()(3) {} 79 30), where the B l () s are the Bernoulli polynomials S () (l) denotes the Stirling numbers of the first ind Most of the other formulas in this article have a similar shape Setting in the above formula for the variable := n N, we obtain n ( ) 3 = 3 n3 n 4π ζ = 3 n3 n 4π ζ n ) ( 3 n 4(n) 79 n 30(n)(n)(n3)(n4), which is the corresponding formula given in our previous paper [8] ( ) l= ( )l (l 3)!! (l) (n)(n) (n) n 4(n)(n) 53 n 640(n)(n)(n3) l (l)! B ls ()

2 Definitions As usual, we denote the floor of by the fractional part of by {} Definition (Pochhammer symbol)[] We define the Pochhammer symbol (or rising factorial function) () by () := ()()(3) ( ) = Γ(), Γ() where Γ() is the gamma function defined by Γ() := 0 e t t dt Definition (Stirling numbers of the first ind)[] Let,l N 0 be two non-negative integers such that l 0 We define the Stirling numbers of the first ind S () (l) as the connecting coefficients in the identity where () is the rising factorial function () = ( ) ( ) l S () (l)l, l=0 Definition 3 (Bernoulli numbers)[] We define the -th Bernoulli number B as the -th coefficient in the generating function relation e = B! C with < π Definition 4 (Euler numbers)[3] We define the sequence of Euler numbers {E } by the generating function identity e e = E! Definition 5 (Bernoulli polynomials)[, 3] We define for n N 0 the n-th Bernoulli polynomial B n () via the following eponential generating function as te t e t = n=0 B n () t n t C with t < π n! Using this epression, we see that the value of the n-th Bernoulli polynomial B n () at the point = 0 is B n (0) = B n, which is the n-th Bernoulli number

3 Definition 6 (Euler polynomials)[3] We define for n N 0 the n-th Euler polynomial E n () via the following eponential generating function as Moreover, we have that [4] e t e t = n=0 E n () t n n! E n (0) = ( n ) B n n n N 0 3 Etended Summation Formulas involving Stirling Series In this section we will prove our summation formulas for various finite sums of the form f() For this, we need the following Lemma 7 (Etended Euler-Maclaurin summation formula)[3, 5] Let f be an analytic function Then for all R, we have that f() = f(t)dt ( )m m! m ( ) B ({})! B m ({t})f (m) (t)dt, f ( ) () m B! f( ) () where B m () is the m-th Bernoulli polynomial {} denotes the fractional part of Therefore, for many functions f we have the asymptotic epansion f() for some constant C C f(t)dtc ( ) B ({})! Lemma 8 (Etended Boole summation formula)[3] Let f be an analytic function Then for all R, we have that ( ) f() = ( ) {} ( )m m! m ( ) E ({})! f () () ( ) t {t} E m ({t})f (m) (t)dt, 3 f ( ) () as, m ( )B f () () ( )!

4 where E m () is the m-th Euler polynomial {} denotes the fractional part of Therefore, for many functions f we have the asymptotic epansion ( ) f() C ( ) {} ( ) E ({})! f () () as, for some constant C C From the above lemma we get by setting := n N the following Corollary 9 (Boole Summation Formula)[3] Let f be an analytic function Then for all n N, we have n m ( ) f() = ( ) n ( ) ( )B f () (n) ( )! ( )m m! the following asymptotic epansion ( ) t {t} E m ({t})f (m) (t)dt, m ( )B f () () ( )! n ( ) f() C ( ) n ( ) ( )B f () (n) as n, ( )! for some constant C C As well as the net ey result found by J Weniger: Lemma 0 (Generalized Weniger transformation)[] For every inverse power series a (), where a () is any function in, the following transformation formula holds a () = ( ) () = Now, we prove as an eample the following ( ) l S () (l)a l() l= ( ) l= ( )l S () (l)a l() ()() () Theorem (Etended summation formulas for the harmonic series) = log()γ B ({}) ( ) l ( ) l= l S() (l)b l({}) ()() () 4

5 We also have that = log()γ ( ) l ( ) l= S () l (l)b l({}) ()() () Proof Applying the etended Euler-Maclaurin summation formula to the function f() :=, we get that log()γ B ({}) Applying now the generalized Weniger transformation to the equivalent series log()γ B ({}) log()γ B ({}) = B ({}) B ({}) ( ), we get the first claimed formula To obtain the second epression, we apply the generalized Weniger transformation to the identity log()γ B ({}) At this point, we want to give an overview on summation formulas obtained with this method: ) Etended summation formulas for the harmonic series: = log()γ B ({}) ( ) l ( ) l= l S() (l)b l({}) ()() () = log()γ ( ) l ( ) l= S () l (l)b l({}) ()() () 5

6 ) Etended summation formulas for the partial sums of ζ(): = ζ() B ({}) ( ) l= ( )l S () (l)b l({}) ()() () = ζ() ( ) l= ( )l S () (l)b l({}) ()() () 3) Etended summation formulas for the partial sums of ζ(3): = ζ(3) 3 B ({}) 3 = ζ(3) 3 ( ) l= ( )l (l)s () (l)b l({}) ()() () ( ) l= ( )l (l)s () (l)b l({}) ()() () 4) Etended summation formulas for the sum of the square roots: = 3 3 4π ζ ( ) 3 B ({}) ( ) l (l 3)!! ( ) l= S () l (l)! (l)b l({}), ()() () = 3 3 4π ζ ( ) 3 B ({}) B ({}) 4 ( ) l (l )!! S () l (l)! ( ) l= (l)b l({}) ()() () = 3 3 4π ζ ( ) 3 ( ) l (l 5)!! ( ) l= S () l l! (l)b l({}) ()() () 6

7 5) Etended summation formulas for the partial sums of ζ( 3/): = π ζ ( ) 5 3 B ({}) 3 3 ( ) l (l 5)!! ( ) l= S () l (l)! (l)b l({}), ()() () = π ζ 3 ( ) 5 3 B ({}) 3 B ({}) B 3({}) 4 4 ( ) l (l )!! ( ) l= S () l (l3)! (l)b l3({}) ()() () = π ζ ( ) ( ) l (l 7)!! ( ) l= S () l l! (l)b l({}) ()() () 6) Etended summation formulas for the partial sums of ζ( 5/): = π 3ζ ( ) 7 5 B ({}) ( ) l (l 7)!! ( ) l= S () l (l)! (l)b l({}), ()() () = π 3ζ 5 4 ( ) 7 5 B ({}) B ({}) 5 B3 ({}) 5B 4({}) 8 64 ( ) l (l )!! ( ) l= S () l (l4)! (l)b l4({}) ()() () = π 3ζ ( ) ( ) l (l 9)!! ( ) l= S () l 3 l! (l)b l({}) ()() () 7

8 7) Etended summation formulas for the sum of the inverse square roots: = ζ ( ) B ({}) ( ) l (l )!! ( ) l= S () l (l)! (l)b l({}) ()() () = ζ ( ) ( ) l (l 3)!! ( ) l= S () l l! (l)b l({}) ()() () 8) Etended summation formulas for the partial sums of ζ(3/): = ζ ( ) 3 B ({}) ( ) l (l)!! ( ) l= S () l (l)! (l)b l({}) ()() () = ζ ( ) 3 ( ) l (l )!! ( ) l= S () l l! (l)b l({}) ()() () 9) Etended summation formulas for the partial sums of ζ(5/): = ζ = ζ ( ) ( ) B ({}) ( ) l (l3)!! ( ) l= S () l (l)! (l)b l({}) ()() () ( ) l (l)!! ( ) l= S () l l! (l)b l({}) ()() () 0) Etended Generalized Faulhaber Formulas: For every positive real number R for every comple number m C\{ }, we have that m = m m ζ( m) m m 8 ( m ) () ( ) l= l S (l)b l({}) ()() ()

9 m = m m ζ( m) m B ({}) m m m = m m ζ( m) m m ( ) m m m m ( m ) () ( ) l= l S (l)b l({}) ()() () ( ) m ( ) B ({}) m ( ) m () ( ) l= l m S ()() () (l)b l m ({}) ) Generalized Faulhaber Formulas: For every comple number m C\{ } every natural number n N, we have that n n m = m nm ζ( m) nm m m = m nm ζ( m) nm nm m n m = m nm ζ( m) m m ( ) m nm m m ( ) ( m ) l= Bl S () l (l) (n)(n) (n) ( m ) ( ) l= Bl S () l (l) (n)(n) (n) ( ) m ( ) B n m ( ) m ( ) l= Bl m S () l m (l) (n)(n) (n) ) Etended convergent versions of Stirling s formula: log() = log() log(π) log()b ({}) ( ) l ( ) l= l(l) S() (l)b l({}) ()() (), 9

10 log() = log() log(π) log()b ({}) B ({}) ( ) l ( ) l= (l)(l) S() (l)b l({}) ()() () that log() = log() log(π) log()b ({}) ( ) l ( ) l= l(l ) S() (l)b l({}) ()() () 3) Etended first logarithmic summation formulas: log() = log() 4 ζ ( ) log()b ({}) log()b ({}) ( ) l ( ) l= l(l)(l) S() (l)b l({}) ()() () log() = log() 4 ( ) log()b ζ ({}) log()b ({}) B 3({}) 6 ( ) l ( ) l= (l)(l)(l3) S() (l)b l3({}) ()() () 4) Etended second logarithmic summation formula: log() = log() γ log() B ({}) log() ( ) l ( ) l= (l)! S() l ()S() (l)b l({}) ()() () ( ) l= l S() (l)b l({}) ()() () 0

11 5) Etended third logarithmic summation formula: log() = ζ () log() ( ) l= log() log() B ({}) l ( l m=0 m l m ) S () (l)b l({}) ()() () ( ) l= S() (l)b l({}) ()() () 6) Etended fourth logarithmic summation formula: log() = log() log() γ π 4 log() log()log(π) log(π) γ log() B ({}) log() B ({}) log() ( ) l ( ) l= (l)! S() l ()S() (l)b l({}) ()() () ( ) l= (l)(l) S() (l)b l({}) ()() () 7) Etended summation formula for partial sums of the Gregory-Leibniz series: ( ) = π 4 ( ) {} Setting := n N, we get that n ( ) = π 4 ( )n ( ) l=0 ( )l l S () (l)e l({}) ()()(3) ( ) ( ) ( ) l l=0 l l ( l )S () (l)b l (n)(n)(n3) (n ) 8) Etended formula for the partial sums of the alternating harmonic series: ( ) = log() ( ) {} ( ) l=0 ( )l S () (l)e l({}) ()() ()

12 Setting := n N, we get n ( ) ( ) l = log()( ) n ( ) l=0 l (l )S () n(n)(n) (n) (l)b l 4 Other Etended Summation Formulas for Finite Sums In this section, we denote by η(s) := ( ) s ) The etended alternating Faulhaber formula: ( ) m = η( m) ( ) {} (m) the Dirichlet eta function m ( ) m ( ) ( ) E ({}) m m N 0 Setting := n N, we get n m ( ) ( ) m = η( m) ( )n m ( ) ( ) B n m m N 0 m ) The etended generalized alternating Faulhaber formula: ( ) m = η( m) ( ) {} m Setting := n N, we get (m) ( ) l=0 (l)( ) m () l S (l)e l({}) ()() () m C n ( ) m = η( m) ( )n n m (m) ( m ) ( ) l=0 l ( l )S () (l)b l (n)(n) (n) m C 3) The Geometric Summation Formula: a = a log(a) a a log(a) l ( ) l= S () l! (l)b l({}) l ()() () a Setting := n N, we get n a = an log(a) a an log(a) l ( ) l= S () l! (l)b ln l (n)(n) (n) a

13 4) The alternating Geometric Summation Formula: ( ) a = a ( ) {} log(a) l a ( ) l=0 S () l! (l)e l({}) l a ()() () Setting := n N, we get n ( ) a = a ( )n a n log(a) l ( ) l=0 (l)! (l )S () (l)b ln l a (n)(n) (n) 5) The Euler-Maclaurin Geometric Summation Formula: a = a log(a) a a ( ) log(a)! B ({}) for e π < a < eπ 6) The Euler-Maclaurin alternating Geometric Summation Formula: ( ) a = (a ) a ( ) {}a a ( ) log(a)! B ({}) for e π < a < eπ 7) The Eponential Geometric Summation Formula: e = e e e ( ) B ({})! R 8) The Self-Counting Summation Formula: Let {a } := {,,,3,3,3,4,4,4,4,} be the self-counting sequence [6, 7] defined by a := Then, we have that a = 8 5 ({ B ({})B }) B ({}) 3 4 ({ 8 B }) ({ 8 8 B }) 4 ({ 8 6 B 3 }) 3

14 9) Slowly convergent summation formula for the sum of the square roots: ( = 3 3 B ({}) FresnelS ) π 3 0) Slowly convergent summation formula for the harmonic series: 5 Conclusion = log()γ CosIntegral(π) This paper presents an overview on some rapidly convergent summation formulas obtained by applying the Weniger transformation [] References [] Ernst Joachim Weniger, Summation of divergent power series by means of factorial series, arxiv: v [mathna], 4 May 00 [] Kevin J McGown Harold R Pars, The generalization of Faulhaber s formula to sums of non-integral powers, J Math Anal Appl 330 (007), [3] Jonathan M Borwein, Neil J Calin, Dante Manna, Euler-Boole summation revisited, The American Mathematical Monthly, Vol 6, No 5 (May, 009), DOI: 0469/ X47090, [4] [5] A V Ustinov, A discrete analog of Euler s summation formula, Mathematical Notes, Vol 7, No 6, 00, [6] [7] [8] Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arxiv: v [mathnt], January 06 4

15 00 Mathematics Subject Classification: Primary 65B5; Secondary B68 Keywords: Rapidly convergent summation formulas for finite sums of functions, etended Stirling summation formulas, etended Euler-Maclaurin summation formula, etended Boole summation formula, generalized Weniger transformation, Stirling numbers of the first ind, Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials 5