Suvarna Garge (Editor)

Gregory coefficients

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers

Contents

that occur in the Maclaurin series expansion of the reciprocal logarithm

z ln ⁡ ( 1 + z ) = 1 + 1 2 z − 1 12 z 2 + 1 24 z 3 − 19 720 z 4 + 3 160 z 5 − 863 60480 z 6 + ⋯ = 1 + ∑ n = 1 ∞ G n z n , | z | < 1 .

Gregory coefficients are alternating Gn = (−1)n−1|Gn| and decreasing in absolute value. These numbers are named after James Gregory who introduced them in 1670 in the numerical integration context. They were subsequently rediscovered by many famous mathematicians and often appear in works of modern authors who do not recognize them.

Computation and representations

The simplest way to compute Gregory coefficients is to use the recurrence formula

G 1 n − G 2 n − 1 + G 3 n − 2 − ⋯ + ( − 1 ) n − 1 G n 1 = 1 n + 1 , n = 2 , 3 , 4 , …

with G1 = 1/2. Gregory coefficients may be also computed explicitly via the following differential

G n = 1 n ! [ d n d z n z ln ⁡ ( 1 + z ) ] z = 0 ,

the integral

G n = 1 n ! ∫ 0 1 x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) d x = ∫ 0 1 ( x n ) d x ,

Schröder's integral formula

G n = ( − 1 ) n − 1 ∫ 0 ∞ d x ( 1 + x ) n ( ln 2 ⁡ x + π 2 ) ,

or the finite summation formula

G n = 1 n ! ∑ ℓ = 1 n s ( n , ℓ ) ℓ + 1 ,

where s(n,ℓ) are the signed Stirling numbers of the first kind.

Bounds and asymptotic behavior

The Gregory coefficients satisfy the bounds

1 6 n ( n − 1 ) < | G n | < 1 6 n , n > 2 ,

given by Johan Steffensen. These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine. In particular,

1 n ln 2 n − 2 n ln 3 n ⩽ | G n | ⩽ 1 n ln 2 n − 2 γ n ln 3 n , n ⩾ 5 .

Asymptotically, at large index n, these numbers behave as

| G n | ∼ 1 n ln 2 ⁡ n , n → ∞ .

More accurate description of Gn at large n may be found in works of Van Veen, Davis, Coffey, Nemes and Blagouchine.

Series with Gregory coefficients

Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include

∑ n = 1 ∞ | G n | = 1 ∑ n = 1 ∞ G n = 1 ln ⁡ 2 − 1 ∑ n = 1 ∞ | G n | n = γ ,

where γ = 0.5772156649... is Euler's constant. These results are very old, and their history may be traced back to the works of Gregorio Fontana and Lorenzo Mascheroni. More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, Alabdulmohsin and some other authors calculated

∑ n = 2 ∞ | G n | n − 1 = − 1 2 + ln ⁡ 2 π 2 − γ 2 ∑ n = 1 ∞ | G n | n + 1 = 1 − ln ⁡ 2.

Alabdulmohsin also gives these identities

| G 1 | + | G 2 | − | G 4 | − | G 5 | + | G 7 | + | G 8 | − | G 10 | − | G 11 | + ⋯ = 3 π | G 2 | + | G 3 | − | G 5 | − | G 6 | + | G 8 | + | G 9 | − | G 11 | − | G 12 | + ⋯ = 2 3 π − 1 | G 1 | − | G 3 | − | G 4 | + | G 6 | + | G 7 | − | G 9 | − | G 10 | + | G 12 | + ⋯ = 1 − 3 π .

Candelperger, Coppo and Young showed that

∑ n = 1 ∞ | G n | ⋅ H n n = π 2 6 − 1 ,

where Hn are the harmonic numbers. Blagouchine provides the following identities

∑ n = 1 ∞ G n n = li ⁡ ( 2 ) − γ ∑ n = 3 ∞ | G n | n − 2 = − 1 8 + ln ⁡ 2 π 12 − ζ ′ ( 2 ) 2 π 2 ∑ n = 4 ∞ | G n | n − 3 = − 1 16 + ln ⁡ 2 π 24 − ζ ′ ( 2 ) 4 π 2 + ζ ( 3 ) 8 π 2 ∑ n = 1 ∞ | G n | n + 2 = 1 2 − 2 ln ⁡ 2 + ln ⁡ 3 ∑ n = 1 ∞ | G n | n + 3 = 1 3 − 5 ln ⁡ 2 + 3 ln ⁡ 3 ∑ n = 1 ∞ | G n | n + k = 1 k + ∑ m = 1 k ( − 1 ) m ( k m ) ln ⁡ ( m + 1 ) , k = 1 , 2 , 3 , … ∑ n = 1 ∞ | G n | n 2 = ∫ 0 1 − li ⁡ ( 1 − x ) + γ + ln ⁡ x x d x ∑ n = 1 ∞ G n n 2 = ∫ 0 1 li ⁡ ( 1 + x ) − γ − ln ⁡ x x d x ,

where li(z) is the integral logarithm and ( k m ) is the binomial coefficient. It is also known that the zeta function, the gamma function, the polygamma functions, the Stieltjes constants and many other special functions and constants may be expressed in terms of infinite series containing these numbers.

Generalizations

Various generalizations are possible for the Gregory coefficients. Many of them may be obtained by modifying the parent generating equation. For example, Van Veen consider

( ln ⁡ ( 1 + z ) z ) s = s ∑ n = 0 ∞ z n n ! K n ( s ) , | z | < 1 ,

and hence

G n = − K n ( − 1 ) n !

Equivalent generalizations were later proposed by Kowalenko and Rubinstein. In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers

( t e t − 1 ) s = ∑ k = 0 ∞ t k k ! B k ( s ) , | t | < 2 π ,

see, so that

G n = − B n ( n − 1 ) ( n − 1 ) n !

Jordan defines polynomials ψn(s) such that

z ( 1 + z ) s ln ⁡ ( 1 + z ) = ∑ n = 0 ∞ z n ψ n ( s ) , | z | < 1 ,

and call them Bernoulli polynomials of the second kind. From the above, it is clear that Gn = ψn(0). Carlitz generalized Jordan's polynomials ψn(s) by introducing polynomials β

( z ln ⁡ ( 1 + z ) ) s ⋅ ( 1 + z ) x = ∑ n = 0 ∞ z n n ! β n ( s ) ( x ) , | z | < 1 ,

and therefore

G n = β n ( 1 ) ( 0 ) n !

Blagouchine introduced numbers Gn(k) such that

G n ( k ) = 1 n ! ∑ ℓ = 1 n s ( n , ℓ ) ℓ + k ,

and studied their asymptotics at large n. Clearly, Gn = Gn(1). A different generalization of the same kind was also proposed by Komatsu

c n ( k ) = ∑ ℓ = 0 n s ( n , ℓ ) ( ℓ + 1 ) k ,

so that Gn = cn(1)/n! Numbers cn(k) are called by the author poly-Cauchy numbers. Coffey defines polynomials

P n + 1 ( y ) = 1 n ! ∫ 0 y x ( 1 − x ) ( 2 − x ) ⋯ ( n − 1 − x ) d x

and therefore |Gn| = Pn+1(1).

References

Gregory coefficients Wikipedia
(Text) CC BY-SA