Vacca-Type Series for Values of the Generalized Euler Constant Function

and its Derivative

Khodabakhsh Hessami Pilehrood^{1} and Tatiana Hessami Pilehrood^{2}

Institute for Studies in Theoretical Physics and Mathematics

Tehran, Iran

                    hessamik@ipm.ir
                  hessamik@gmail.com
                    hessamit@ipm.ir
                  hessamit@gmail.com

Abstract

We generalize well-known Catalan-type integrals for Euler’s constant to values of the generalized Euler constant function and its derivatives. Using generating functions appearing in these integral representations, we give new Vacca and Ramanujan-type series for values of the generalized Euler constant function and Addison-type series for values of the generalized Euler constant function and its derivative. As a consequence, we get base-B rational series for log \frac{4}{π} , \frac{G}{π} (where G is Catalan’s constant), \frac{ζ′(2)}{π^{2}} and also for logarithms of the Somos and Glaisher-Kinkelin constants.

1 Introduction

J. Sondow [24] proved the following two formulas:

γ = ∑_{n=1}^{∞} \frac{N_{1,2}(n) + N_{0,2}(n)}{2n(2n + 1)},  (1)

1Author’s current address: Mathematics Department, Faculty of Basic Sciences, Shahrekord University, Shahrekord, P.O. Box 115, Iran. Research was in part supported by a grant from IPM (No. 87110018).

2Author’s current address: Mathematics Department, Faculty of Basic Sciences, Shahrekord University, Shahrekord, P.O. Box 115, Iran. Research was in part supported by a grant from IPM (No. 87110019).

log \frac{4}{π} = ∑_{n=1}^{∞} \frac{N_{1,2}(n) − N_{0,2}(n)}{2n(2n + 1)},  (2)

where γ is Euler’s constant and N_{i,2}(n) is the number of i’s in the binary expansion of n (see sequences \underline{A000120} and \underline{A023416} in Sloane’s database [23]). The series (1) is equivalent to the well-known Vacca series [28]

γ = ∑_{n=1}^{∞}(−1)^{n} \frac{⌊log_{2} n⌋}{n} = ∑_{n=1}^{∞}(−1)^{n} \frac{N_{1,2}(⌊\frac{n}{2}⌋) + N_{0,2}(⌊\frac{n}{2}⌋)}{n}  (3)

and both series (1) and (3) may be derived from Catalan’s integral [8]

γ = ∫_{0}^{1} \frac{1}{1 + x} ∑_{n=1}^{∞} x^{2^{n}−1} dx.  (4)

To see this it suffices to note that

G(x) = \frac{1}{1 − x} ∑_{n=0}^{∞} x^{2^{n}} = ∑_{n=1}^{∞}(N_{1,2}(n) + N_{0,2}(n))x^{n}

is a generating function of the sequence N_{1,2}(n)+N_{0,2}(n), (see \underline{A070939}), which is the binary length of n, rewrite (4) as

γ = ∫_{0}^{1}(1 − x)\frac{G(x^{2})}{x} dx

and integrate the power series termwise. In view of the equality

1 = ∫_{0}^{1} ∑_{n=1}^{∞} x^{2^{n}−1} dx,

which is easily verified by termwise integration, (4) is equivalent to the formula

γ = 1 − ∫_{0}^{1} \frac{1}{1 + x} ∑_{n=1}^{∞} x^{2^{n}} dx  (5)

obtained independently by Ramanujan (see [5, Corollary 2.3]). Catalan’s integral (5) gives the following rational series for γ :

γ = 1 − ∫_{0}^{1}(1 − x)G(x^{2}) dx = 1 − ∑_{n=1}^{∞} \frac{N_{1,2}(n) + N_{0,2}(n)}{(2n + 1)(2n + 2)}.  (6)

Averaging (1), (6) and (4), (5), respectively, we get Addison’s series for γ [1]

γ = \frac{1}{2} + ∑_{n=1}^{∞} \frac{N_{1,2}(n) + N_{0,2}(n)}{2n(2n + 1)(2n + 2)}

and its corresponding integral

γ = \frac{1}{2} + \frac{1}{2} ∫_{0}^{1} \frac{1 − x}{1 + x} ∑_{n=1}^{∞} x^{2^{n}−1} dx,  (7)

respectively. Integrals (5), (4) were generalized to an arbitrary integer base B > 1 by S. Ramanujan and by B. C. Berndt and D. C. Bowman (see [5]):

γ = 1 − ∫_{0}^{1} (\frac{1}{1 − x} − \frac{Bx^{B−1}}{1 − x^{B}} ) ∑_{n=1}^{∞} x^{B^{n}} dx  (Ramanujan),  (8)

γ = ∫_{0}^{1} (\frac{B}{1 − x^{B}} − \frac{1}{1 − x}) ∑_{n=1}^{∞} x^{B^{n}−1} dx  (Berndt-Bowman).  (9)

Formula (9) implies the generalized Vacca series for γ (see [5, Theorem 2.6]) proposed by L. Carlitz [7]:

γ = ∑_{n=1}^{∞} \frac{ε(n)}{n}⌊log_{B} n⌋,  (10)

where

ε(n) = \begin{cases}B − 1, & if B divides n; \\ −1, & otherwise; \end{cases}  (11)

and the averaging integral of (8) and (9) produces the generalized Addison series for γ found by Sondow [24]:

γ = \frac{1}{2} + ∑_{n=1}^{∞} \frac{⌊log_{B} Bn⌋P_{B}(n)}{Bn(Bn + 1) ⋯ (Bn + B)},  (12)

where P_{B}(x) is a polynomial of degree B − 2 defined by

P_{B}(x) = (Bx + 1)(Bx + 2) ⋯ (Bx + B − 1) ∑_{m=1}^{B−1} \frac{m(B − m)}{Bx + m}.  (13)

In this paper, we consider the generalized Euler constant function

γ_{a,b}(z) = ∑_{n=0}^{∞} (\frac{1}{an + b} − log (\frac{an + b + 1}{an + b})) z^{n},  a, b ∈ ℕ,  |z| ≤ 1,  (14)

which is related to the constants in (1), (2) as γ_{1,1}(1) = γ, γ_{1,1}(−1) = log \frac{4}{π}. Basic properties of a special case of this function, γ_{1,1}(z), were studied earlier in [25, 14]. In Section 2, we show that γ_{a,b}(z) admits an analytic continuation to the domain ℂ \ [1, +∞) in terms of the Lerch transcendent. In Sections 3–4, we generalize Catalan-type integrals (8), (9) to values of the generalized Euler constant function and its derivatives. Using generating functions appearing in these integral representations, we give new Vacca- and Ramanujan-type series for values of γ_{a,b}(z) and Addison-type series for values of γ_{a,b}(z) and its derivative. In Section 5, we get base-B rational series for log \frac{4}{π}, \frac{G}{π}, (where G is Catalan’s constant), \frac{ζ′(2)}{π^{2}} and also for logarithms of the Somos and Glaisher-Kinkelin constants. We also mention a connection of our approach to summation of series of the form

∑_{n=1}^{∞} N_{ω,B}(n)Q(n, B) and ∑_{n=1}^{∞} \frac{N_{ω,B}(n)P_{B}(n)}{Bn(Bn + 1) ⋯ (Bn + B)},

where Q(n, B) is a rational function of B and n

Q(n, B) = \frac{1}{Bn(Bn + 1)} + \frac{2}{Bn(Bn + 2)} + ⋯ + \frac{B − 1}{Bn(Bn + B − 1)},  (15)

and N_{ω,B}(n) is the number of occurrences of a word ω over the alphabet {0, 1, …, B − 1} in the B-ary expansion of n, considered in [2]. Moreover, we answer some questions posed in [2] concerning possible generalizations of the series (1) and (2) to any integer base B > 1. Note that in the above notation, the generalized Vacca series (10) can be written as follows:

γ = ∑_{k=1}^{∞} L_{B}(k)Q(k, B),  (16)

where L_{B}(k) := ⌊log_{B} Bk⌋ = ∑_{α=0}^{B−1} N_{α,B}(k) is the B-ary length of k. Indeed, representing n = Bk + r, 0 ≤ r ≤ B − 1 and summing in (10) over k ≥ 1 and 0 ≤ r ≤ B − 1 we get

γ = ∑_{k=1}^{∞}⌊log_{B} Bk⌋ (\frac{B − 1}{Bk} − \frac{1}{Bk + 1} − ⋯ − \frac{1}{Bk + B − 1}) = ∑_{k=1}^{∞}⌊log_{B} Bk⌋Q(k, B).

Using the same notation, the generalized Addison series (12) gives another base-B expansion of Euler’s constant

γ = \frac{1}{2} + ∑_{n=1}^{∞} \frac{L_{B}(n)P_{B}(n)}{Bn(Bn + 1) ⋯ (Bn + B)} = \frac{1}{2} + ∑_{n=1}^{∞} L_{B}(n) (Q(n, B) − \frac{B − 1}{2Bn(n + 1)}) (17)

which converges faster than (16) to γ. Here we used the fact that

∑_{n=1}^{∞} ∑_{α=0}^{B−1} \frac{N_{α,B}(n)}{n(n + 1)} = \frac{B}{B − 1},

which can be easily checked by [3, Section 3]. On the other hand,

\begin{matrix}Q & (n, B) − \frac{B − 1}{2Bn(n + 1)} = \frac{1}{2} ∑_{m=1}^{B−1} (\frac{1}{Bn} − \frac{2}{Bn + m} + \frac{1}{Bn + B} ) \\ = \frac{1}{Bn(Bn + B)} ∑_{m=1}^{B−1} (2m − B + \frac{2m(B − m)}{Bn + m}) = \frac{P_{B}(n)}{Bn(Bn + 1) ⋯ (Bn + B)}. \end{matrix}

Finally, we give a brief description of some other generalized Euler constants that have appeared in the literature in Section 6.

2 Analytic continuation

We consider the generalized Euler constant function γ_{a,b}(z) defined in (14), where a, b are positive real numbers, z ∈ ℂ, and the series converges when |z| ≤ 1. We show that γ_{a,b}(z) admits an analytic continuation to the domain ℂ\[1, +∞). The following theorem is a slight modification of [25, Theorem 3].

Theorem 1. Let a, b be positive real numbers, z ∈ ℂ, |z| ≤ 1. Then

γ_{a,b}(z) = ∫_{0}^{1} ∫_{0}^{1} \frac{(xy)^{b−1}(1 − x)}{(1 − zx^{a}y^{a})(− log xy)} dxdy = ∫_{0}^{1} \frac{x^{b−1}(1 − x)}{1 − zx^{a}} (\frac{1}{1 − x} + \frac{1}{log x}) dx. (18)

The integrals converge for all z ∈ ℂ \ (1, +∞) and give the analytic continuation of the generalized Euler constant function γ_{a,b}(z) for z ∈ ℂ \ [1, +∞).

Proof. Denoting the double integral in (18) by I(z) and for |z| ≤ 1, expanding (1 − zx^{a}y^{a})^{−1} in a geometric series we have

I(z) = ∑_{k=0}^{∞} z^{k} ∫_{0}^{1} ∫_{0}^{1} \frac{(xy)^{ak+b−1}(1 − x)}{(− log xy)} dxdy

= ∑_{k=0}^{∞} z^{k} ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{+∞}(xy)^{t+ak+b−1}(1 − x) dxdydt

= ∑_{k=0}^{∞} z^{k} ∫_{0}^{+∞} (\frac{1}{(t + ak + b)^{2}} − (\frac{1}{t + ak + b} − \frac{1}{t + ak + b + 1})) dt = γ_{a,b}(z).

On the other hand, making the change of variables u = x^{a}, v = y^{a} in the double integral we get

I(z) = \frac{1}{a} ∫_{0}^{1} ∫_{0}^{1} \frac{(uv)^{\frac{b}{a} −1}(1 − u^{\frac{1}{a}})}{(1 − zuv)(− log uv)} dudv.

Now by [12, Corollary 3.3], for z ∈ ℂ \ [1, +∞) we have

I(z) = \frac{1}{a}Φ(z, 1, \frac{b}{a}) − \frac{∂Φ}{∂s}(z, 0, \frac{b}{a}) + \frac{∂Φ}{∂s}(z, 0, \frac{b + 1}{a}),

where Φ(z, s, u) is the Lerch transcendent, a holomorphic function in z and s, for z ∈ ℂ \ [1, +∞) and all complex s (see [12, Lemma 2.2]), which is the analytic continuation of the series

Φ(z, s, u) = ∑_{n=0}^{∞} \frac{z^{n}}{(n + u)^{s}} , u > 0.

To prove the second equality in (18), make the change of variables X = xy, Y = y and integrate with respect to Y.

Corollary 2. Let a, b be positive real numbers, l ∈ ℕ, z ∈ ℂ \ [1, +∞). Then for the l-th derivative we have

γ_{a,b}^{(l)}(z) = ∫_{0}^{1} ∫_{0}^{1} \frac{(xy)^{al+b−1}(x − 1)}{(1 − zx^{a}y^{a})^{l+1} log xy} dxdy = ∫_{0}^{1} \frac{x^{la+b−1}(1 − x)}{(1 − zx^{a})^{l+1}} (\frac{1}{1 − x} + \frac{1}{log x}) dx.

From Corollary 2, [12, Cor.3.3, 3.8, 3.9] and [2, Lemma 4] we get

Corollary 3. Let a, b be positive real numbers, z ∈ ℂ\[1, +∞). Then the following equalities hold:

γ_{a,b}(1) = log Γ(\frac{b + 1}{a}) − log Γ(\frac{b}{a}) − \frac{1}{a}ψ(\frac{b}{a}),

γ_{a,b}(z) = \frac{1}{a}Φ(z, 1, \frac{b}{a}) − \frac{∂Φ}{∂s}(z, 0, \frac{b}{a}) + \frac{∂Φ}{∂s}(z, 0, \frac{b + 1}{a}),

\begin{aligned}γ′_{a,b}(z) = & − \frac{b}{a^{2}} Φ(z, 1, \frac{b}{a} + 1) + \frac{1}{a(1 − z)} + \frac{b}{a} \frac{∂Φ}{∂s}(z, 0, \frac{b}{a} + 1) − \frac{∂Φ}{∂s}(z, −1, \frac{b}{a} + 1)− \\ & \frac{b + 1}{a} \frac{∂Φ}{∂s}(z, 0, \frac{b + 1}{a} + 1) + \frac{∂Φ}{∂s}(z, −1, \frac{b + 1}{a} + 1), \end{aligned}

where Φ(z, s, u) is the Lerch transcendent and ψ(x) = \frac{d}{dx} log Γ(x) is the logarithmic derivative of the gamma function.

3 Catalan-type integrals for γ_{a,b}^{(l)}(z).

Berndt and Bowman [5] demonstrated that for x > 0 and any integer B > 1,

\frac{1}{1 − x} + \frac{1}{log x} = ∑_{k=1}^{∞} \frac{(B − 1) + (B − 2)x^{\frac{1}{B^{k}}} + (B − 3)x^{\frac{2}{B^{k}}} + ⋯ + x^{\frac{B−2}{B^{k}}}}{B^{k}(1 + x^{\frac{1}{B^{k}}} + x^{\frac{2}{B^{k}}} + ⋯ + x^{\frac{B−1}{B^{k}}})} .  (19)

The special cases B = 2, 3 of this equality can be found in Ramanujan’s third note book [21, p. 364]. Using this key formula we prove the following generalization of the integral (9).

Theorem 4. Let a, b, B be positive integers with B > 1, l a non-negative integer. If either z ∈ ℂ \ [1, +∞) and l ≥ 1, or z ∈ ℂ \ (1, +∞) and l = 0, then

γ_{a,b}^{(l)}(z) = ∫_{0}^{1} (\frac{B}{1 − x^{B}} − \frac{1}{1 − x}) F_{l}(z, x) dx  (20)

where

F_{l}(z, x) = ∑_{k=1}^{∞} \frac{x^{(b+al)B^{k}−1}(1 − x^{B^{k}})}{(1 − zx^{aB^{k}})^{l+1}}.  (21)

Proof. First we note that the series of variable x on the right-hand side of (19) converges uniformly on [0, 1], since the absolute value of its general term does not exceed \frac{B−1}{2B^{k−1}} . Then for l ≥ 0, multiplying both sides of (19) by \frac{x^{la+b−1}(1−x)}{(1−zx^{a})^{l+1}} and integrating over 0 ≤ x ≤ 1 we get

γ_{a,b}^{(l)}(z) = ∑_{k=1}^{∞} ∫_{0}^{1} \frac{x^{la+b−1}(1 − x)}{(1 − zx^{a})^{l+1}} · \frac{(B − 1) + (B − 2)x^{\frac{1}{B^{k}}} + ⋯ + x^{\frac{B−2}{B^{k}}}}{B^{k}(1 + x^{\frac{1}{B^{k}}} + x^{\frac{2}{B^{k}}} + ⋯ + x^{\frac{B−1}{B^{k}}})} dx.

Replacing x by x^{B^{k}} in each integral we find

\begin{aligned}γ_{a,b}^{(l)}(z) & = ∑_{k=1}^{∞} ∫_{0}^{1} \frac{x^{(la+b)B^{k}−1}(1 − x^{B^{k}})}{(1 − zx^{aB^{k}})^{l+1}} · \frac{(B − 1) + (B − 2)x + ⋯ + x^{B−2}}{1 + x + x^{2} + ⋯ + x^{B−1}} dx \\ & = ∫_{0}^{1} (\frac{B}{1 − x^{B}} − \frac{1}{1 − x}) F_{l}(z, x) dx, \end{aligned}

as required.

From Theorem 4 we readily get a generalization of Ramanujan’s integral.

Corollary 5. Let a, b, B be positive integers with B > 1, l a non-negative integer. If either z ∈ ℂ \ [1, +∞) and l ≥ 1, or z ∈ ℂ \ (1, +∞) and l = 0, then

γ_{a,b}^{(l)}(z) = ∫_{0}^{1} \frac{x^{b+al−1}(1 − x)}{(1 − zx^{a})^{l+1}} dx + ∫_{0}^{1} (\frac{Bx^{B}}{1 − x^{B}} − \frac{x}{1 − x}) F_{l}(z, x) dx.  (22)

Proof. First we note that the series (21), considered as a sum of functions of the variable x converges uniformly on [0, 1 − ε] for any ε > 0. Then integrating termwise we have

∫_{0}^{1−ε} F_{l}(z, x) dx = ∑_{k=1}^{∞} ∫_{0}^{1−ε} \frac{x^{(b+al)B^{k}−1}(1 − x^{B^{k}})}{(1 − zx^{aB^{k}})^{l+1}} dx.

Making the change of variable y = x^{B^{k}} in each integral we get

∫_{0}^{1−ε} F_{l}(z, x) dx = ∑_{k=1}^{∞} \frac{1}{B^{k}} ∫_{0}^{(1−ε)^{B^{k}}} \frac{y^{b+al−1}(1 − y)}{(1 − zy^{a})^{l+1}} dy.

Since the last series, considered as a series in the variable ε, converges uniformly on [0, 1], letting ε tend to zero we get

∫_{0}^{1} F_{l}(z, x) dx = \frac{1}{B − 1} ∫_{0}^{1} \frac{y^{b+al−1}(1 − y)}{(1 − zy^{a})^{l+1}} dy.  (23)

Now from (20) and (23) it follows that

γ_{a,b}^{(l)}(z) − ∫_{0}^{1} \frac{y^{b+al−1}(1 − y)}{(1 − zy^{a})^{l+1}} dy = ∫_{0}^{1} (\frac{Bx^{B}}{1 − x^{B}} − \frac{x}{1 − x}) F_{l}(z, x) dx,

and the proof is complete.

Averaging the formulas (20) and (22), we get the following generalization of the integral (7).

Corollary 6. Let a, b, B be positive integers with B > 1, l a non-negative integer. If either z ∈ ℂ \ [1, +∞) and l ≥ 1, or z ∈ ℂ \ (1, +∞) and l = 0, then

γ_{a,b}^{(l)}(z) = \frac{1}{2} ∫_{0}^{1} \frac{x^{b+al−1}(1 − x)}{(1 − zx^{a})^{l+1}} dx + \frac{1}{2} ∫_{0}^{1} (\frac{B(1 + x^{B})}{1 − x^{B}} − \frac{1 + x}{1 − x}) F_{l}(z, x) dx.

4 Vacca-type series for γ_{a,b}(z) and γ′_{a,b}(z).

Theorem 7. Let a, b, B be positive integers with B > 1, z ∈ ℂ, |z| ≤ 1. Then for the generalized Euler constant function γ_{a,b}(z), the following expansion is valid:

γ_{a,b}(z) = ∑_{k=1}^{∞} a_{k}Q(k, B) = ∑_{k=1}^{∞} a_{⌊ \frac{k}{B} ⌋} \frac{ε(k)}{k},

where Q(k, B) is a rational function given by (15), {a_{k}}_{k=0}^{∞} is a sequence defined by the generating function

G(z, x) = \frac{1}{1 − x} ∑_{k=0}^{∞} \frac{x^{bB^{k}}(1 − x^{B^{k}})}{1 − zx^{aB^{k}}} = ∑_{k=0}^{∞} a_{k}x^{k}  (24)

and ε(k) is defined in (11).

Proof. For l = 0, rewrite (20) in the form

γ_{a,b}(z) = ∫_{0}^{1} \frac{1 − x^{B}}{x} (\frac{B}{1 − x^{B}} − \frac{1}{1 − x}) G(z, x^{B}) dx

where G(z, x) is defined in (24). Then, since a_{0} = 0, we have

γ_{a,b}(z) = ∫_{0}^{1}(B − 1 − x − x^{2} − ⋯ − x^{B−1}) ∑_{k=1}^{∞} a_{k}x^{Bk−1} dx.  (25)

Expanding G(z, x) in a power series of x,

G(z, x) = ∑_{k=0}^{∞} ∑_{m=0}^{∞} z^{m}x^{(am+b)B^{k}}(1 + x + ⋯ + x^{B^{k}−1}),

we see that a_{k} = O(ln_{B} k). Therefore, by termwise integration in (25), which can be easily justified by the same way as in the proof of Corollary 5, we get

\begin{aligned}γ_{a,b}(z) & = ∑_{k=1}^{∞} a_{k} ∫_{0}^{1}[(x^{Bk−1} − x^{Bk}) + (x^{Bk−1} − x^{Bk+1}) + ⋯ + (x^{Bk−1} − x^{Bk+B−2})] dx \\ & = ∑_{k=1}^{∞} a_{k}Q(k, B). \end{aligned}

Theorem 8. Let a, b, B be positive integers with B > 1, z ∈ ℂ, |z| ≤ 1. Then for the generalized Euler constant function, the following expansion is valid:

γ_{a,b}(z) = ∫_{0}^{1} \frac{x^{b−1}(1 − x)}{1 − zx^{a}} dx − ∑_{k=1}^{∞} a_{k}Q̃(k, B),

where

\begin{aligned}Q̃(k, B) & = \frac{B − 1}{Bk(k + 1)} − Q(k, B) \\ & = \frac{B − 1}{(Bk + B)(Bk + 1)} + \frac{B − 2}{(Bk + B)(Bk + 2)} + ⋯ + \frac{1}{(Bk + B)(Bk + B − 1)} \end{aligned}

and the sequence {a_{k}}_{k=1}^{∞} is defined in Theorem 7.

Proof. From Corollary 5 with l = 0, using the same method as in the proof of Theorem 7, we get

∫_{0}^{1} (\frac{Bx^{B}}{1 − x^{B}} − \frac{x}{1 − x}) F_{0}(z, x) = ∫_{0}^{1} \frac{1 − x^{B}}{x} (\frac{Bx^{B}}{1 − x^{B}} − \frac{x}{1 − x}) G(z, x^{B}) dx

= ∫_{0}^{1}(Bx^{B−1} − (1 + x + ⋯ + x^{B−1})) ∑_{k=1}^{∞} a_{k}x^{Bk} dx

= ∑_{k=1}^{∞} a_{k} ∫_{0}^{1}[(x^{Bk+B−1} − x^{Bk+B−2}) + ⋯ + (x^{Bk+B−1} − x^{Bk+1}) + (x^{Bk+B−1} − x^{Bk})] dx

= − ∑_{k=1}^{∞} a_{k}Q̃(k, B).

Theorem 9. Let a, b, B be positive integers with B > 1, z ∈ ℂ, |z| ≤ 1. Then for the generalized Euler constant function γ_{a,b}(z) and its derivative, the following expansion is valid:

γ_{a,b}^{(l)}(z) = \frac{1}{2} ∫_{0}^{1} \frac{x^{b+al−1}(1 − x)}{(1 − zx^{a})^{l+1}} dx + ∑_{k=1}^{∞} a_{k,l} \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)}, l = 0, 1,

where P_{B}(k) is a polynomial of degree B − 2 given by (13), z ≠ 1 if l = 1, and the sequence {a_{k,l}}_{k=0}^{∞} is defined by the generating function

G_{l}(z, x) = \frac{1}{1 − x} ∑_{k=0}^{∞} \frac{x^{(b+al)B^{k}}(1 − x^{B^{k}})}{(1 − zx^{aB^{k}})^{l+1}} = ∑_{k=0}^{∞} a_{k,l}x^{k},  l = 0, 1.  (26)

Proof. Expanding G_{l}(z, x) in a power series of x,

G_{l}(z, x) = ∑_{k=0}^{∞} ∑_{m=0}^{∞} \binom{m + l}{l}z^{m}x^{(b+al+am)B^{k}}(1 + x + x^{2} + ⋯ + x^{B^{k}−1}),

we see that a_{k,l} = O(k^{l} ln_{B} k). Therefore, for l = 0, 1, by termwise integration we get

∫_{0}^{1} (\frac{B(1 + x^{B})}{1 − x^{B}} − \frac{1 + x}{1 − x}) F_{l}(z, x)dx = ∫_{0}^{1} \frac{1 − x^{B}}{x} (\frac{B(1 + x^{B})}{1 − x^{B}} − \frac{1 + x}{1 − x}) G_{l}(z, x^{B})dx

= ∫_{0}^{1}[(B − 1) − 2x − 2x^{2} − ⋯ − 2x^{B−1} + (B − 1)x^{B}] ∑_{k=1}^{∞} a_{k,l}x^{Bk−1} dx

= ∑_{k=1}^{∞} a_{k,l} (\frac{B − 1}{Bk} − \frac{2}{Bk + 1} − \frac{2}{Bk + 2} − ⋯ − \frac{2}{Bk + B − 1} + \frac{B − 1}{Bk + B} )

= 2 ∑_{k=1}^{∞} a_{k,l} \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},

where P_{B}(k) is defined in (13) and the last series converges since \frac{P_{B}(k)}{Bk(Bk+1)···(Bk+B)} = O(k^{−3}). Now our theorem easily follows from Corollary 6.

5 Examples of rational series

It is easily seen that the generating function (26) satisfies the following functional equation:

G_{l}(z, x) − \frac{1 − x^{B}}{1 − x} G_{l}(z, x^{B}) = \frac{x^{b+al}}{(1 − zx^{a})^{l+1}} ,  (27)

which is equivalent to the following identity for series:

∑_{k=0}^{∞} a_{k,l}x^{k} − (1 + x + ⋯ + x^{B−1}) ∑_{k=0}^{∞} a_{k,l}x^{Bk} = ∑_{k=l}^{∞} \binom{k}{l}z^{k−l}x^{ak+b}.

Comparing coefficients of powers of x we get an alternative definition of the sequence {a_{k,l}}_{k=0}^{∞} by means of the recursion

a_{0,l} = a_{1,l} = ⋯ = a_{al+b−1,l} = 0

and for k ≥ al + b,

a_{k,l} = \begin{cases}a_{⌊ \frac{k}{B} ⌋,l}, & if k ≢ b (mod a); \\ a_{⌊ \frac{k}{B} ⌋,l} + \binom{(k−b)/a}{l}z^{\frac{k−b}{a} −l}, & if k ≡ b (mod a). \end{cases}  (28)

On the other hand, in view of Corollary 3, γ_{a,b}(z) and γ′_{a,b}(z) can be explicitly expressed in terms of the Lerch transcendent, ψ-function and logarithm of the gamma function. This allows us to sum the series in Theorems 7–9 in terms of these functions.

Example 10. Suppose that ω is a non-empty word over the alphabet {0, 1, …, B − 1}. Then obviously ω is uniquely defined by its length |ω| and its size v_{B}(ω) which is the value of ω when interpreted as an integer in base B. Let N_{ω,B}(k) be the number of (possibly overlapping) occurrences of the block ω in the B-ary expansion of k. Note that for every B and ω, N_{ω,B}(0) = 0, since the B-ary expansion of zero is the empty word. If the word ω begins with 0, but v_{B}(ω) ≠ 0, then in computing N_{ω,B}(k) we assume that the B-ary expansion of k starts with an arbitrary long prefix of 0’s. If v_{B}(ω) = 0 we take for k the usual shortest B-ary expansion of k.

Now we consider equation (27) with l = 0, z = 1

G(1, x) − \frac{1 − x^{B}}{1 − x} G(1, x^{B}) = \frac{x^{b}}{1 − x^{a}}  (29)

and for a given non-empty word ω, set a = B^{|ω|} in (29) and

b = \begin{cases}B^{|ω|}, & if v_{B}(ω) = 0; \\ v_{B}(ω), & if v_{B}(ω) ≠ 0. \end{cases}

Then by (28), it is easily seen that a_{k} := a_{k,0} = N_{ω,B}(k), k = 1, 2, …, and by Theorem 7, we get another proof of the following statement (see [2, Sections 3, 4.2]).

Corollary 11. Let ω be a non-empty word over the alphabet {0, 1, …, B − 1}. Then

∑_{k=1}^{∞} N_{ω,B}(k)Q(k, B) = \begin{cases}γ_{B^{|ω|},v_{B}(ω)}(1), & if v_{B}(ω) ≠ 0; \\ γ_{B^{|ω|},B^{|ω}|}(1), & if v_{B}(ω) = 0. \end{cases}

By Corollary 3, the right-hand side of the last equality can be calculated explicitly and we have

∑_{k=1}^{∞} N_{ω,B}(k)Q(k, B) = \begin{cases}log Γ (\frac{v_{B}(ω)+1}{B^{|ω|}}) − log Γ (\frac{v_{B}(ω)}{B^{|ω|}}) − \frac{1}{B^{|ω|}} ψ (\frac{v_{B}(ω)}{B^{|ω|}}) , & if v_{B}(ω) ≠ 0; \\ log Γ (\frac{1}{B^{|ω|}} ) + \frac{γ}{B^{|ω|}} − |ω| log B, & if v_{B}(ω) = 0. \end{cases}

(30)

Corollary 12. Let ω be a non-empty word over the alphabet {0, 1, …, B − 1}. Then

\begin{matrix}∑_{k=1}^{∞} & \frac{N_{ω,B}(k)P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)} \\ = \begin{cases}γ_{B^{|ω|},v_{B}(ω)}(1) − \frac{1}{2B^{|ω|}} (ψ(\frac{v_{B}(ω)+1}{B^{|ω|}}) − ψ(\frac{v_{B}(ω)}{B^{|ω|}})) , & if v_{B}(ω) ≠ 0; \\ γ_{B^{|ω|},B^{|ω}|}(1) − \frac{1}{2B^{|ω|}} ψ(\frac{1}{B^{|ω|}} ) − \frac{γ}{2B^{|ω|}} − \frac{1}{2}, & if v_{B}(ω) = 0. \end{cases} \end{matrix}

Proof. The required statement easily follows from Theorem 9, Corollary 11 and the equality

∫_{0}^{1} \frac{x^{b−1}(1 − x)}{1 − x^{a}} dx = ∑_{k=0}^{∞} (\frac{1}{ak + b} − \frac{1}{ak + b + 1}) = \frac{1}{a} (ψ(\frac{b + 1}{a}) − ψ(\frac{b}{a})) .

From Theorem 7, (27) and (28) with a = 1, l = 0 we have Corollary 13. Let b, B be positive integers with B > 1, z ∈ ℂ, |z| ≤ 1. Then

γ_{1,b}(z) = ∑_{k=1}^{∞} a_{k}Q(k, B) = ∑_{k=1}^{∞} a_{⌊ \frac{k}{B} ⌋} \frac{ε(k)}{k},

where a_{0} = a_{1} = ⋯ = a_{b−1} = 0, a_{k} = a_{⌊ \frac{k}{B} ⌋} + z^{k−b}, k ≥ b.

Similarly, from Theorem 9 we have Corollary 14. Let b, B be positive integers with B > 1, z ∈ ℂ, |z| ≤ 1. Then

γ_{1,b}(z) = \frac{1}{2} ∑_{k=0}^{∞} \frac{z^{k}}{(k + b)(k + b + 1)} + ∑_{k=1}^{∞} a_{k} \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},

where a_{0} = a_{1} = ⋯ = a_{b−1} = 0, a_{k} = a_{⌊ \frac{k}{B} ⌋} + z^{k−b}, k ≥ b. Example 15. If in Corollary 13 we take z = 1, then we get that a_{k} is equal to the B-ary length of ⌊\frac{k}{b}⌋, i. e.,

a_{k} = ∑_{α=0}^{B−1} N_{α,B} (⌊\frac{k}{b} ⌋) = L_{B} (⌊\frac{k}{b} ⌋) .

On the other hand,

γ_{1,b}(1) = log b − ψ(b) = log b − ∑_{k=1}^{b−1} \frac{1}{k} + γ

and hence we get

log b − ψ(b) = ∑_{k=1}^{∞} L_{B} (⌊\frac{k}{b} ⌋) Q(k, B).  (31)

If b = 1, formula (31) gives (16). If b > 1, then from (31) and (16) we get

log b = ∑_{k=1}^{b−1} \frac{1}{k} + ∑_{k=1}^{∞} (L_{B} (⌊\frac{k}{b} ⌋) − L_{B}(k)) Q(k, B),  (32)

which is equivalent to [5, Theorem 2.8]. Similarly, from Corollary 14 we obtain (17) and

log b = ∑_{k=1}^{b−1} \frac{1}{k} − \frac{b − 1}{2b} + ∑_{k=1}^{∞} \frac{(L_{B}(⌊\frac{k}{b}⌋) − L_{B}(k)) P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)}.  (33)

Example 16. Using the fact that for any integer B > 1,

L_{B} (⌊\frac{k}{B} ⌋) − L_{B}(k) = −1,

from (30), (16) and (32) we get the following rational series for log Γ(1/B) :

log Γ (\frac{1}{B} ) = ∑_{k=1}^{B−1} \frac{1}{k} + ∑_{k=1}^{∞}(N_{0,B}(k) − \frac{1}{B} L_{B}(k) − 1)Q(k, B).

Example 17. Substituting b = 1, z = −1 in Corollary 13 we get the generalized Vacca series for log \frac{4}{π}.

Corollary 18. Let B ∈ ℕ, B > 1. Then

log \frac{4}{π} = ∑_{k=1}^{∞} a_{k}Q(k, B) = ∑_{k=1}^{∞} a_{⌊ \frac{k}{B} ⌋} \frac{ε(k)}{k},

where

a_{0} = 0,  a_{k} = a_{⌊ \frac{k}{B} ⌋} + (−1)^{k−1}, k ≥ 1.  (34)

In particular, if B is even, then

log \frac{4}{π} = ∑_{k=1}^{∞}(N_{odd,B}(k) − N_{even,B}(k))Q(k, B) = ∑_{k=1}^{∞} \frac{(N_{odd,B}(⌊\frac{k}{B} ⌋) − N_{even,B}(⌊\frac{k}{B} ⌋))}{k}ε(k), (35)

where N_{odd,B}(k) (respectively N_{even,B}(k)) is the number of occurrences of the odd (respectively even) digits in the B-ary expansion of k.

Proof. To prove (35), we notice that if B is even, then the sequence ã_{k} := N_{odd,B}(k) − N_{even,B}(k) satisfies recurrence (34).

Substituting b = 1, z = −1 in Corollary 14 with the help of (33) we get the generalized Addison series for log \frac{4}{π}.

Corollary 19. Let B > 1 be a positive integer. Then

log \frac{4}{π} = \frac{1}{4} + ∑_{k=1}^{∞} \frac{(L_{B}(⌊\frac{k}{2}⌋) − L_{B}(k) + a_{k}) P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},

where the sequence a_{k} is defined in Corollary 18. In particular, if B is even, then

log \frac{4}{π} = \frac{1}{4} + ∑_{k=1}^{∞} \frac{(L_{B}(⌊\frac{k}{2}⌋) − 2N_{even,B}(k)) P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)}.

Example 20. For t > 1, the generalized Somos constant σ_{t} is defined by

σ_{t} = \sqrt[t]{1\sqrt[t]{2\sqrt[t]{3 ⋯}}} = 1^{1/t}2^{1/t^{2}}3^{1/t^{3}} ⋯ = ∏_{n=1}^{∞} n^{1/t^{n}}

(see [25, Section 3]). In view of the relation [25, Theorem 8]

γ_{1,1} (\frac{1}{t}) = t log \frac{t}{(t − 1)σ_{t}^{t−1}} ,  (36)

by Corollary 13 and formula (32) we get Corollary 21. Let B ∈ ℕ, B > 1, t ∈ ℝ, t > 1. Then

log σ_{t} = \frac{1}{(t − 1)^{2}} + \frac{1}{t − 1} ∑_{k=1}^{∞} (L_{B}(⌊\frac{k}{t} ⌋) − L_{B}(⌊\frac{k}{t − 1}⌋) − \frac{a_{k}}{t} ) Q(k, B),

where a_{0} = 0, a_{k} = a_{⌊ \frac{k}{B} ⌋} + t^{1−k}, k ≥ 1.

In particular, setting B = t = 2 we get the following rational series for Somos’s quadratic recurrence constant:

log σ_{2} = 1 − \frac{1}{2} ∑_{k=1}^{∞} \frac{b_{k}}{2k(2k + 1)},

where b_{1} = 3, b_{k} = b_{⌊ \frac{k}{2} ⌋} + \frac{1}{2^{k−1}} , k ≥ 2.

From (36), (33) and Theorem 9 we find

Corollary 22. Let B ∈ ℕ, B > 1, t ∈ ℝ, t > 1. Then

\begin{matrix}log σ_{t} = \frac{3t − 1}{4t(t − 1)^{2}} \\ + \frac{t + 1}{2(t − 1)} ∑_{k=1}^{∞} (L_{B}(⌊\frac{k}{t} ⌋) − L_{B}(⌊\frac{k}{t − 1}⌋) − \frac{2a_{k}}{t(t + 1)}) \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)}, \end{matrix}

where the sequence a_{k} is defined in Corollary 21.

In particular, if B = t = 2 we get

log σ_{2} = \frac{5}{8} − \frac{1}{2} ∑_{k=1}^{∞} \frac{c_{k}}{2k(2k + 1)(2k + 2)},

where c_{1} = 4, c_{k} = c_{⌊ \frac{k}{2} ⌋} + \frac{1}{2^{k−1}} , k ≥ 2.

Example 23. The Glaisher-Kinkelin constant is defined by the limit [11, p.135]

A := lim_{n→∞} \frac{1^{2}2^{2} ⋯ n^{n}}{n^{\frac{n^{2}+n}{2} + \frac{1}{12}} e^{− \frac{n^{2}}{4}}} = 1.28242712 · · · .

Its connection to the generalized Euler constant function γ_{a,b}(z) is given by the formula [25, Corollary 4]

γ′_{1,1}(−1) = log \frac{2^{11/6}A^{6}}{π^{3/2}e} .  (37)

By Theorem 9, since

∫_{0}^{1} \frac{x(1 − x)}{(1 + x)^{2}} dx = 3 log 2 − 2,

we have

log A = \frac{4}{9} log 2 − \frac{1}{4} log \frac{4}{π} + \frac{1}{6} ∑_{k=1}^{∞} a_{k,1} \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},

where the sequence a_{k,1} is defined by the generating function (26) with a = b = l = 1, z = −1, or using the recursion (28):

a_{0,1} = a_{1,1} = 0, a_{k,1} = a_{⌊ \frac{k}{B} ⌋,1} + (−1)^{k}(k − 1), k ≥ 2.

Now by Corollary 19 and (33) we get

Corollary 24. Let B > 1 be a positive integer. Then

log A = \frac{13}{48} − \frac{1}{36} ∑_{k=1}^{∞} (7L_{B}(k) − 7L_{B}(⌊\frac{k}{2}⌋) + b_{k}) \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},

where b_{0} = 0, b_{k} = b_{⌊ \frac{k}{B} ⌋} + (−1)^{k−1}(6k + 3), k ≥ 1.

In particular, if B = 2 we get

log A = \frac{13}{48} − \frac{1}{36} ∑_{k=1}^{∞} \frac{c_{k}}{2k(2k + 1)(2k + 2)},

where c_{1} = 16, c_{k} = c_{⌊ \frac{k}{2} ⌋} + (−1)^{k−1}(6k + 3), k ≥ 2.

Using the formula expressing \frac{ζ′(2)}{π^{2}} in terms of the Glaisher-Kinkelin constant [11, p. 135],

log A = −\frac{ζ′(2)}{π^{2}} + \frac{log 2π + γ}{12} ,

by Corollaries 14, 19 and 24, we get

Corollary 25. Let B > 1 be a positive integer. Then

\frac{ζ′(2)}{π^{2}} = −\frac{1}{16} + \frac{1}{36} ∑_{k=1}^{∞} (4L_{B}(k) − L_{B}(⌊\frac{k}{2}⌋) + c_{k}) \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},

where c_{0} = 0, c_{k} = c_{⌊ \frac{k}{B} ⌋} + (−1)^{k−1}6k, k ≥ 1.

Example 26. First we evaluate γ_{2,1}^{(l)}(−1) for l = 0, 1. From Corollaries 2, 3 and [12, Examples 3.9, 3.15] we have

γ_{2,1}(−1) = ∫_{0}^{1} ∫_{0}^{1} \frac{(x − 1) dxdy}{(1 + x^{2}y^{2}) log xy} = \frac{π}{4} − 2 log Γ(\frac{1}{4}) + log \sqrt{2π^{3}}

and

\begin{aligned}γ′_{2,1}(−1) = & − \frac{1}{4}Φ(−1, 1, 3/2) + \frac{1}{2}Φ(−1, 0, 3/2) + \frac{1}{2} \frac{∂Φ}{∂s}(−1, 0, 3/2) \\ & − \frac{∂Φ}{∂s}(−1, −1, 3/2) − \frac{∂Φ}{∂s}(−1, 0, 2) + \frac{∂Φ}{∂s}(−1, −1, 2). \end{aligned}

The last expression can be evaluated explicitly (see [12, Section 2]) and we get

γ′_{2,1}(−1) = \frac{G}{π} + \frac{π}{8} − log Γ(\frac{1}{4}) − 3 log A + log π + \frac{1}{3} log 2,

or

\frac{G}{π} = γ′_{2,1}(−1) − \frac{1}{2}γ_{2,1}(−1) + \frac{1}{4} log \frac{4}{π} + 3 log A − \frac{7}{12} log 2.  (38)

On the other hand, by Theorem 9 and (28) we have

γ_{2,1}(−1) = \frac{π}{8} − \frac{1}{4} log 2 + ∑_{k=1}^{∞} a_{k,0} \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},  (39)

where a_{0,0} = 0, a_{2k,0} = a_{⌊ \frac{2k}{B} ⌋,0}, k ≥ 1, a_{2k+1,0} = a_{⌊ \frac{2k+1}{B} ⌋,0} + (−1)^{k}, k ≥ 0, and

γ′_{2,1}(−1) = \frac{π}{16} − \frac{1}{4} log 2 + ∑_{k=1}^{∞} a_{k,1} \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},  (40)

where a_{0,1} = 0, a_{2k,1} = a_{⌊ \frac{2k}{B} ⌋,1}, k ≥ 1, a_{2k+1,1} = a_{⌊ \frac{2k+1}{B} ⌋,1} + (−1)^{k−1}k, k ≥ 0. Now from (38) – (40), (33) and Corollary 19 we get the following expansion for G/π.

Corollary 27. Let B > 1 be a positive integer. Then

\frac{G}{π} = \frac{11}{32} + ∑_{k=1}^{∞} (\frac{1}{8}L_{B}(⌊\frac{k}{2}⌋) − \frac{1}{8}L_{B}(k) + c_{k}) \frac{P_{B}(k)}{Bk(Bk + 1) ⋯ (Bk + B)},

where c_{0} = 0, c_{2k} = c_{⌊ \frac{2k}{B} ⌋} + k, k ≥ 1, c_{2k+1} = c_{⌊ \frac{2k+1}{B} ⌋} + \frac{(−1)^{k−1}−1}{2}(2k + 1), k ≥ 0.

In particular, if B = 2 we get

\frac{G}{π} = \frac{11}{32} + ∑_{k=1}^{∞} \frac{b_{k}}{2k(2k + 1)(2k + 2)},

where b_{1} = −\frac{9}{8}, b_{2k} = b_{k} + k, b_{2k+1} = b_{k} + \frac{(−1)^{k−1}−1}{2}(2k + 1), k ≥ 1.

6 Other generalized Euler constants

The purpose of this section is to draw attention to different generalizations of Euler’s constant for which many interesting results remain to be discovered.

The simplest way to generalize Euler’s constant

γ = lim_{n→∞} (∑_{k=1}^{n} \frac{1}{k} − log n) ,  (41)

which is related to the digamma function by the equality γ = −ψ(1), is to consider for 0 < α ≤ 1,

γ(α) = lim_{n→∞} (∑_{k=1}^{n} \frac{1}{k + α} − log n) = lim_{n→∞} (∑_{k=1}^{n} \frac{1}{k + α} − log(n + α)) .

Tasaka [27] proved that γ(α) = −ψ(α). Its connection to the generalized Euler constant function γ_{a,b}(z) is given by the formula

γ(α) + log α = γ_{1,α}(1).

Briggs [6] and Lehmer [20] studied the analog of γ corresponding to the arithmetical progression of positive integers r, r + m, r + 2m, …, (r ≤ m) :

γ(r, m) = lim_{n→∞} (H(n, r, m) − \frac{1}{m} log n) ,

where H(n, r, m) = _{∑_{0<k≤n,} ; k≡r(mod m)} \frac{1}{k}. Since H(n, r, m) = ∑_{0≤k≤(n−r)/m} \frac{1}{r+mk}, it is easily seen that

mγ(r, m) = γ(r/m) − log m = γ_{1,r/m}(1) − log r.

Diamond and Ford [9] studied a family {γ(P)} of generalized Euler constants arising from the sum of reciprocals of integers sieved by finite sets of primes P. More precisely, if P represents a finite set of primes, let

1_{P}(n) := \begin{cases}1, & if gcd (n, ∏_{p∈P} p) = 1; \\ 0, & otherwise; \end{cases} and δ_{P} := lim_{x→∞} \frac{1}{x} ∑_{n≤x} 1_{P}(n).

A simple argument shows that δ_{P} = ∏_{p∈P}(1 − 1/p) and that the generalized Euler constant

γ(P) := lim_{x→∞} (∑_{n≤x} \frac{1_{P}(n)}{n} − δ_{P} log x)

exists. Its connection to Euler’s constant is given by the formula [9]

γ(P) = ∏_{p∈P} (1 − \frac{1}{p}) \{γ + ∑_{p∈P} \frac{log p}{p − 1}\} .

Another generalization of the Euler constant is connected with the well-known limit involving the Riemann zeta function:

γ = lim_{s→1} (ζ(s) − \frac{1}{s − 1}) .  (42)

Expanding the Riemann zeta function into Laurent series in a neighborhood of its simple pole at s = 1 gives

ζ(s) = \frac{1}{s − 1} + ∑_{k=0}^{∞} \frac{(−1)^{k}}{k!} γ_{k}(s − 1)^{k}.

Stieltjes [26] pointed out that the coefficients γ_{k} can be expressed as

γ_{k} = lim_{n→∞} (∑_{j=1}^{n} \frac{log^{k} j}{j} − \frac{log^{k+1} n}{k + 1}) ,  k = 0, 1, 2, ….  (43)

(In the case k = 0, the first summand requires evaluation of 0^{0}, which is taken to be 1.) The coefficients γ_{k} are usually called Stieltjes, or generalized Euler, constants (see [10, 11, 18]. In particular, the zero’th constant γ_{0} = γ is the Euler constant.

Hardy [13] gave an analog of the Vacca series (3) for γ_{1} containing logarithmic coefficients:

γ_{1} = ∑_{k=1}^{∞}(−1)^{k} \frac{log(k)⌊log_{2}(k)⌋}{k} − \frac{log 2}{2} ∑_{k=1}^{∞}(−1)^{k} \frac{⌊log_{2}(2k)⌋⌊log_{2}(k)⌋}{k},

and Kluyver [16] presented more such series for higher-order constants.

The analog of γ_{k} corresponding to the arithmetical progression r, r + m, r + 2m, … was studied by Knopfmacher [17], Kanemitsu [15], and Dilcher [10]:

γ_{k}(r, m) = lim_{n→∞}(_{∑_{0<j≤n} ; j≡r (mod m)} \frac{log^{k} j}{j} − \frac{1}{m} \frac{log^{k+1} n}{k + 1}).

Another extension of γ_{k} can be derived from the Laurent series expansion of the Hurwitz zeta function:

ζ(s, α) := ∑_{n=0}^{∞} \frac{1}{(n + α)^{s}} = \frac{1}{s − 1} + ∑_{k=0}^{∞} \frac{(−1)^{k}γ_{k}(α)}{k!}(s − 1)^{k}.

Here 0 < α ≤ 1. Since ζ(s, 1) = ζ(s), we have γ_{k}(1) = γ_{k}. Berndt [4] showed that

γ_{k}(α) = lim_{n→∞} (∑_{j=0}^{n} \frac{log^{k}(j + α)}{(j + α)} − \frac{log^{k+1}(n + α)}{k + 1}) ,

which is equivalent to (43) when α = 1. If k = 0 and α = r/m, r, m ∈ ℕ, r ≤ m, then

γ_{0}(r/m) = mγ(r, m) + log m = γ_{1,r/m}(1) − log(r/m) = γ(r/m) = −ψ(r/m).

Recently, Lampret [19] considered the zeta-generalized Euler constant function

Υ(s) := ∑_{j=1}^{∞} (\frac{1}{j^{s}} − ∫_{j}^{j+1} \frac{dx}{x^{s}})  (44)

and its alternating version

Υ^{∗}(s) := ∑_{j=1}^{∞}(−1)^{j+1} (\frac{1}{j^{s}} − ∫_{j}^{j+1} \frac{dx}{x^{s}})

defined for s ≥ 0. The name of the function Υ(s) comes from the fact that Υ(1) = γ and that the series ∑_{j=1}^{∞} 1/j^{s} defines the Riemann zeta function. Moreover, it is easily seen that Υ(1) = γ_{1,1}(1) and Υ^{∗}(1) = γ_{1,1}(−1). In [19] it was shown that Υ(s) is infinitely differentiable on ℝ^{+} and its k-th derivative Υ^{(k)}(s) can be obtained by termwise k-times differentiation of the series (44):

Υ^{(k)}(s) = (−1)^{k} ∑_{j=1}^{∞} (\frac{log^{k} j}{j^{s}} − ∫_{j}^{j+1} \frac{log^{k} x}{x^{s}} dx) .  (45)

Setting s = 1 in (45) we get the following relation between the zeta-generalized Euler constant function and Stieltjes constants (43):

Υ^{(k)}(1) = (−1)^{k}γ_{k}.

The formula (41), as well as (44), can be further generalized to

γ_{f} = lim_{n→∞} (∑_{k=1}^{n} f(k) − ∫_{1}^{n} f(x) dx)

for some arbitrary positive decreasing function f (see [22]). For example, f_{n}(x) = \frac{log^{n} x}{x} gives rise to the Stieltjes constants, and f_{s}(x) = x^{−s} gives γ_{f_{s}} = \frac{(s−1)ζ(s)−1}{s−1}, where again the limit (42) appears.

There are other generalizations including a two-dimensional version of Euler’s constant and a lattice sum form. For a survey of further results and an extended bibliography, see [11, Sections 1.5, 1.10, 2.21, 7.2].

7 Acknowledgements

We thank the Max Planck Institute for Mathematics at Bonn where this research was carried out. Special gratitude is due to professor B. C. Berndt for providing paper [5]. We thank the referee for useful comments, which improved the presentation of the paper.

References

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2010 Mathematics Subject Classification: Primary 11Y60; Secondary 65B10, 40A05, 05A15. Keywords: Euler constant, series summation, generating function, Lerch transcendent, generalized Somos constants, Glaisher-Kinkelin constant.

(Concerned with sequences \underline{A000120}, \underline{A023416}, and \underline{A070939}.)

Received October 19 2009; revised version received June 27 2010. Published in Journal of Integer Sequences, July 9 2010.

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