bluuebunny/crossref_metadata_2025_split huggingface dataset:
Subordination-preserving integral operators
Authors: Miller Sanford S. , Mocanu Petru T. , Reade Maxwell O. 
Published: June 2010

Let

β
\beta

and

γ
\gamma

be complex numbers and let

H
H

be the space of functions regular in the unit disc. Subordination of functions

f
f

,

g
∈
H

g \in H

is denoted by

f
≺
g

f \prec g

. Let

K
⊂
H

K \subset H

and let the operator

A
:
K
→
H

A:K \to H

be defined by

F
=
A
(
f
)

F = A(f)

, where
\[

F
(
z
)
=

[

1

z
γ

∫
0
z

f
β

(
t
)

t

γ
−
1

d
t

]

1

/

β

.

F(z) = {\left [ {\frac {1} {{{z^\gamma }}}\int _0^z {{f^\beta }(t){t^{\gamma - 1}}dt} } \right ]^{1/\beta }}.

\]
The authors determine conditions under which
\[

f
≺
g
⇒
A
(
f
)
≺
A
(
g
)
,

f \prec g \Rightarrow A(f) \prec A(g),

\]
and then use this result to obtain new distortion theorems for some classes of regular functions.
The dual Weierstrass-Laguerre transform
Authors: Haimo Deborah Tepper 
Published: June 2010

An inversion algorithm is derived for the dual Weierstrass-Laguerre transform

∫
0
∞

g
α

(
x
,
y
;
1
)
φ
(
y
)

y
α

e

−
y

/

(
α
+
1
)
d
y

\int _0^\infty {{g_\alpha }(x,y;1)\varphi (y){y^\alpha }{e^{ - y}}/(\alpha + 1)dy}

, where the function

g
α

(
x
,
y
,
t
)

{g_\alpha }(x,y,t)

is associated with the source solution of the Laguerre differential heat equation

x

u

x
x

(
x
,
t
)
=
(
α
+
1
−
x
)

u
x

(
x
,
t
)
=

u
t

(
x
,
t
)

x{u_{xx}}(x,t) = (\alpha + 1 - x){u_x}(x,t) = {u_t}(x,t)

. Correspondingly, sufficient conditions are established for a function to be represented by a Weierstrass-Laguerre Stieltjes transform

∫
0
∞

g
α

(
x
,
y
;
1
)

d
β
(
y
)

\int _0^\infty {{g_\alpha }(x,y;1)\;d\beta (y)}

of a nondecreasing function

β
\beta

.
Long time asymptotics of the Korteweg-de Vries equation
Authors: Venakides Stephanos 
Published: June 2010

We study the long time evolution of the solution to the Kortewegde Vries equation with initial data

υ
(
x
)

\upsilon (x)

which satisfy
\[

lim

x
→
−
∞

υ
(
x
)
=
−
1
,

lim

x
→
+
∞

υ
(
x
)
=
0

\lim \limits _{x \to - \infty } \upsilon (x) = - 1,\qquad \lim \limits _{x \to + \infty } \upsilon (x) = 0

\]
We show that as

t
→
∞

t \to \infty

the step emits a wavetrain of solitons which asymptotically have twice the amplitude of the initial step. We derive a lower bound of the number of solitons separated at time

t
t

for

t
t

large.
Real hypersurfaces and complex submanifolds in complex projective space
Authors: Kimura Makoto 
Published: June 2010

Let

M
M

be a real hypersurface in

P
n

(

C

)

{P^n}({\mathbf {C}})

be the complex structure and

ξ
\xi

denote a unit normal vector field on

M
M

. We show that

M
M

is (an open subset of) a homogeneous hypersurface if and only if

M
M

has constant principal curvatures and

J
ξ

J\xi

is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically,

P
m

(

C

)

{P^m}({\mathbf {C}})

(totally geodesic),

Q
n

,

P
1

(

C

)
×

P
n

(

C

)
,
S
U
(
5
)

/

S
(
U
(
2
)
×
U
(
3
)
)

{Q^n},{P^1}({\mathbf {C}}) \times {P^n}({\mathbf {C}}),SU(5)/S(U(2) \times U(3))

and

S
O
(
10
)

/

U
(
5
)

SO(10)/U(5)

are the only complex submanifolds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.
On the behavior near the crest of waves of extreme form
Authors: Amick C. J. , Fraenkel L. E. 
Published: June 2010

The angle

ϕ
\phi

which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle

2
π

/

3

2\pi /3

. In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion
\[

(
∗
)

ϕ
(
s
)
=

π
6

+

∑

n
=
1

k

a
n

s

μ
n

+
o
(

s

μ
k

)

as

s
↓
0

( * )\qquad \phi (s) = \frac {\pi } {6} + \sum \limits _{n = 1}^k {{a_n}{s^{{\mu _n}}} + o({s^{{\mu _k}}})} \quad {\text {as}}\,s \downarrow 0

\]
, to arbitrary order; the coordinate

s
s

is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents

μ
n

{\mu _n}

are transcendental numbers. We are unable to evaluate the coefficients

a
n

{a_n}

explicitly, but define some in terms of global properties of

ϕ
\phi

, and the others in terms of earlier coefficients. It is proved in [8] that

a
1

>
0

{a_1} > 0

, and follows here that

a
2

>
0

{a_2} > 0

. The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of

n
n

.
Two-dimensional nonlinear boundary value problems for elliptic equations
Authors: Lieberman Gary M. 
Published: June 2010

Boundary regularity of solutions of the fully nonlinear boundary value problem
\[

F
(
x
,
u
,
D
u
,

D
2

u
)
=
0

in

Ω
,

G
(
x
,
u
,
D
u
)
=
0

on

∂
Ω

F(x,u,Du,{D^2}u) = 0\quad {\text {in}}\;\Omega ,\qquad G(x,u,Du) = 0\quad {\text {on}}\;\partial \Omega

\]
is discussed for two-dimensional domains

Ω
\Omega

. The function

F
F

is assumed uniformly elliptic and

G
G

is assumed to depend (in a nonvacuous manner) on

D
u

Du

. Continuity estimates are proved for first and second derivatives of

u
u

under weak hypotheses for smoothness of

F
F

,

G
G

, and

Ω
\Omega

.
Sharp distortion theorems for quasiconformal mappings
Authors: Anderson G. D. , Vamanamurthy M. K. , Vuorinen M. 
Published: June 2010

Continuing their earlier work on distortion theory, the authors prove some dimension-free distortion theorems for

K
K

-quasiconformal mappings in

R
n

{R^n}

. For example, one of the present results is the following sharp variant of the Schwarz lemma: If

f
f

is a

K
K

-quasiconformal self-mapping of the unit ball

B
n

{B^n}

,

n
⩾
2

n \geqslant 2

, with

f
(
0
)
=
0

f(0) = 0

, then

4

1
−

K
2

|

x

|

K

⩽

|

f
(
x
)

|

⩽

4

1
−
1

/

K
2

|

x

|

1

/

K

{4^{1 - {K^2}}}|x{|^K} \leqslant |f(x)| \leqslant {4^{1 - 1/{K^2}}}|x{|^{1/K}}

for all

x
x

in

B
n

{B^n}

.
Weighted norm inequalities for potential operators
Authors: Schechter Martin 
Published: June 2010

We give sufficient conditions for inequalities of the form\[(∫(∫G(x−y)f(y)dμ(y))qdω(x))1/q⩽C(∫|f(y)|pdν(y))1/p{\left ( {\int {{{\left ( {\int {G(x - y)f(y)\,d\mu (y)} } \right )}^q}\,d\omega (x)} } \right )^{1/q}}\, \leqslant C{\left ( {\int {|f(y){|^p}d\nu (y)} } \right )^{1/p}}\]to hold for measurable functionsff. We determine the dependence of the constantCCon the measuresμ\mu,ν\nu,ω\omegaand give some applications.
Explicit formula for weighted scalar nonlinear hyperbolic conservation laws
Authors: LeFloch Philippe , Nédélec Jean-Claude 
Published: June 2010

We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form
\[

∂

∂
t

(
r
u
)
+

∂

∂
x

(
r
f
(
u
)
)
=
0
,

\frac {\partial } {{\partial t}}(ru) + \frac {\partial } {{\partial x}}(rf(u)) = 0,

\]
with initial data and boundary condition. The scalar function

u
=
u
(
x
,

t
)

u = u(x,\,t)

,

x
>
0

x > 0

,

t
>
0

t > 0

, is the unknown, the function

f
=
f
(
u
)

f = f(u)

is assumed to be strictly convex with inf

f
(
⋅
)
=
0

f( \cdot ) = 0

and the weight function

r
=
r
(
x
)

r = r(x)

,

x
>
0

x > 0

, to be positive (for example,

r
(
x
)
=

x
α

r(x) = {x^\alpha }

, with an arbitrary real

α
\alpha

). We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux

r
(
⋅
)
f
(
u
(
⋅
,
⋅
)
)

r( \cdot )f(u( \cdot , \cdot ))

at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to B. L. Keyfitz.
Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms
Authors: Salzer Herbert E. 
Published: June 2010

In finding

f
(
t
)

f(t)

, the inverse Laplace transform of

F
(
p
)

F(p)

, where (1)

f
(
t
)
=
(
1

/

2
π
j
)

∫

c
−
j
∞

c
+
j
∞

e

p
t

F
(
p
)
d
p

f(t) = (1/2\pi j)\int _{c - j\infty }^{c + j\infty }{e^{{p^t}}}F(p)dp

, the function

F
(
p
)

F(p)

may be either known only numerically or too complicated for evaluating

f
(
t
)

f(t)

by Cauchy’s theorem. When

F
(
p
)

F(p)

behaves like a polynomial without a constant term, in the variable

1

/

p

1/p

, along (

c
−
j
∞
,

c
+
j
∞

c - j\infty ,\;c + j\infty

), one may find

f
(
t
)

f(t)

numerically using new quadrature formulas (analogous to those employing the zeros of the Laguerre polynomials in the direct Laplace transform). Suitable choice of

p
i

{p_i}

yields an

n
n

-point quadrature formula that is exact when

ρ

2
n

{\rho _{2n}}

is any arbitrary polynomial of the (

2
n

2n

)th degree in

x
≡
1

/

p

x \equiv 1/p

without a constant term, namely: (2)

(
1

/

2
π
j
)

∫

c
−
j
∞

c
+
j
∞

e
p

ρ

2
n

(
1

/

p
)
d
p
=

∑

i
=
1

n

A
i

(
n
)

ρ

2
n

(
1

/

p
i

)

(1/2\pi j)\int _{c - j\infty }^{c + j\infty } {{e^p}} {\rho _{2n}}(1/p)dp = {\sum \limits _{i = 1}^n {{A_i}} ^{(n)}}{\rho _{2n}}(1/{p_i})

. In (2),

x
i

≡
1

/

p
i

{x_i} \equiv 1/{p_i}

are the zeros of the orthogonal polynomials

P
n

(
x
)
≡

∏

i
=
1

n

(
x
−

x
i

)

{P_n}(x) \equiv \prod \limits _{i = 1}^n {(x - {x_i})}

where (3)

(
1

/

2
π
j
)

∫

c
−
j
∞

c
+
j
∞

e
p

(
1

/

p
)

p
n

(
1

/

p
)

(
1

/

p

)
i

d
p
=
0
,

i
=
0
,
1
,
⋯
,
n
−
1

(1/2\pi j)\int _{c - j\infty }^{c + j\infty } {{e^p}} (1/p){p_n}(1/p){(1/p)^i}dp = 0,\;i = 0,1, \cdots ,n - 1

and

A
i

(
n
)

{A_i}^{(n)}

correspond to the Christoffel numbers. The normalization

P
n

(
1

/

p
)
≡
(
4
n
−
2
)
(
4
n
−
6
)
⋯
6

p
n

(
1

/

p
)
,
n
≥
2

{P_n}(1/p) \equiv (4n - 2)(4n - 6) \cdots 6{p_n}(1/p),n \geq 2

, produces all integral coefficients.

P
n

(
1

/

p
)

{P_n}(1/p)

is proven to be

(
−
1

)
n

e

−
p

p
n

d
n

(

e
p

/

p
n

)

/

d

p
n

{( - 1)^n}{e^{ - p}}{p^n}{d^n}({e^p}/{p^n})/d{p^n}

. The normalization factor is proved, in three different ways, to be given by (4)

(
1

/

2
π
j
)

∫

c
−
j
∞

c
+
j
∞

e
p

(
1

/

p
)

[

P
n

(
1

/

p
)

]
2

d
p
=

1
2

(
−
1

)
n

(1/2\pi j)\int _{c - j\infty }^{c + j\infty } {{e^p}} (1/p){[{P_n}(1/p)]^2}dp = \tfrac {1}{2}{( - 1)^n}

. Proofs are given for the recurrence formula (5)

(
2
n
−
3
)

P
n

(
x
)
=
[
(
4
n
−
2
)
(
2
n
−
3
)
x
+
2
]

P

n
−
1

(
x
)
+
(
2
n
−
1
)

P

n
−
2

(
x
)

(2n - 3){P_n}(x) = [(4n - 2)(2n - 3)x + 2]{P_{n - 1}}(x) + (2n - 1){P_{n - 2}}(x)

, for

n
≥
3

n \geq 3

, and the differential equation (6)

x
2

P
n

(
x
)
+
(
x
−
1
)

P
n

′

(
x
)
−

n
2

P
n

(
x
)
=
0

{x^2}{P_n}^{}(x) + (x - 1){P_n}’(x) - {n^2}{P_n}(x) = 0

. The quantities

p
i

(
n
)

,
1

/

p
i

(
n
)

{p_i}^{(n)},1/{p_i}^{(n)}

and

A
i

(
n
)

{A_i}^{(n)}

were computed, mostly to 6S – 8S, for

i
=
1
(
1
)
n
,
n
=
1
(
1
)
8

i = 1(1)n,n = 1(1)8

.
Complex zeros of cylinder functions
Authors: Döring Boro 
Published: June 2010

All complex zeros of the cylinder functions

Y
n

(
z
)
,

H
n

(
1
)

(
z
)

for

n
=
0

{Y_n}(z),{H_n}^{(1)}(z){\text { for }}n = 0

and

1
1

,

|

z

|

>
158

|z| > 158

,

|

arg
⁡
z

|

≦
π

|\arg z| \leqq \pi

and some complex zeros small in modulus of

Y
n

(
z
)

{Y_n}(z)

,

H
n

(
1
)

(
z
)

{H_n}^{(1)}(z)

for

n
=
2

n = 2

,

3
3

,

4
4

,

5
5

,

15
15

and

|

arg
⁡
z

|

≦
π

|\arg z| \leqq \pi

are tabulated accurate to

10
D

10D

. The methods (being partly new) for computing these zeros are described.
On the numerical solution of a differential-difference equation arising in analytic number theory
Authors: van de Lune J. , Wattel E. 
Published: June 2010

In the January 1962 issue of this Journal R. Bellman and B. Kotkin published a short paper under the same title as this one (cf. [1]). In that paper Bellman and Kotkin presented some of their results concerning the numerical computation of the continuous function

y
(
x
)

y(x)

, defined by
\[

y
(
x
)
=
1
(
0
≦
x
≦
1
)

y
′

(
x
)
=
−

1
x

⋅
y
(
x
−
1
)
(
x
>
1
)

\begin {gathered} y(x) = 1(0 \leqq x \leqq 1) \hfill \\ y’ (x) = - \frac {1} {x} \cdot y(x - 1)(x > 1) \hfill \\ \end {gathered}

\]
Tables of

y
(
x
)

y(x)

were given for

x
=
1
(
0.0625
)
6

x = 1(0.0625)6

and

x
=
6
(
1
)
20

x = 6(1)20

. In the process of extending these tables beyond

x
=
20

x = 20

we discovered that the second table was rather inaccurate for all values of

x
>
9

x > 9

. Bellman and Kotkin found, for example, that

y
(
20
)
=
0.149
⋅

10

−
8

y(20) = 0.149 \cdot {10^{ - 8}}

, whereas the actual value of

y
(
20
)

y(20)

can be shown to be smaller than

10

−
20

{10^{ - 20}}

. Moreover, in view of the method used by Bellman and Kotkin, one may expect that it would be quite time consuming to compute

y
(
x
)

y(x)

for values of

x
x

up to say

x
=
1
,
000

x = 1,000

. In this paper we describe a different method which enables us to compute

y
(
x
)

y(x)

for values of

x
x

up to about “as far as one would like."
Summation of a slowly convergent series arising in antenna study
Authors: Den Chi Fu 
Published: June 2010

An equivalent series for the slowly convergent series
\[

∑

n
=
1

∞

[

∫

−
π

/

2

π

/

2

cos

α

θ
cos
⁡
(
n
∈
sin
⁡
θ
)

]

2

/

n

\sum \limits _{n = 1}^\infty {\left [ {\smallint _{ - \pi /2}^{\pi /2}{{\cos }^\alpha }\theta \cos (n \in \sin \theta )} \right ]} {}^2/n

\]
which arises in antenna theory is obtained. The new form is found to consist of two rapidly convergent series for small

∈
\in

.
Chebyshev type quadrature formulas
Authors: Kahaner David K. 
Published: June 2010

Quadrature formulas of the form
\[

∫

−
1

1

f
(
x
)
d
x
≈

2
n

∑

i
=
1

n

f
(

x
i

(
n
)

)

\int _{ - 1}^1 {f(x)dx \approx \frac {2} {n}\sum \limits _{i = 1}^n {f({x_i}^{(n)})} }

\]
are associated with the name of Chebyshev. Various constraints may be posed on the formula to determine the nodes

x
i

(
n
)

{x_i}^{(n)}

. Classically the formula is required to integrate

n
n

th degree polynomials exactly. For

n
=
8

n = 8

and

n
≧
10

n \geqq 10

this leads to some complex nodes. In this note we point out a simple way of determining the nodes so that the formula is exact for polynomials of degree less than

n
n

. For

n
=
8

n = 8

,

10
10

and

11
11

we compare our results with others obtained by minimizing the

l
2

{l^2}

-norm of the deviations of the first

n
+
1

n + 1

monomials from their moments and point out an error in one of these latter calculations.
An integral analogue of Taylor’s series and its use in computing Fourier transforms
Authors: Osler Thomas J. 
Published: June 2010

In this paper, an integral analogue of Taylor’s series
\[

f
(
z
)
=

∫

−
∞

∞

D
ω

f
(

z
0

)

(
z
−

z
0

)
ω

/

Γ
(
ω
+
1
)
d
ω

f(z) = \int _{ - \infty }^\infty {{D^\omega }f({z_0})} {(z - {z_0})^\omega }/\Gamma (\omega + 1)d\omega

\]
is discussed.

D
ω

f
(
z
)

{D^\omega }f(z)

is a fractional derivative of order

ω
\omega

. Extensions of this integral are also given, one of which is an integral analogue of Lagrange’s expansion. These integrals are shown to be generalizations of the Fourier integral theorem. Several special cases of these integrals are computed, and a table of Fourier transforms emerges.
Minimum norm differentiation formulas with improved roundoff error bounds
Authors: Kahaner David K. 
Published: June 2010

Numerical differentiation formulas of the form

Σ

i
=
1

N

w
i

f
(

x
i

)
≈

f

(
m
)

(
a
)
,
α
≦

x
i

≦
β

\Sigma _{i = 1}^N{w_i}f({x_i}) \approx {f^{(m)}}(a),\alpha \leqq {x_i} \leqq \beta

, are considered. The roundoff error of such formulas is bounded by a value proportional to

Σ

i
=
1

N

|

w
i

|

\Sigma _{i = 1}^N|{w_i}|

. We consider formulas that have minimum norm

Σ

i
=
1

N

w
i
2

\Sigma _{i = 1}^Nw_i^2

and converge to

f

(
m
)

(
a
)

{f^{(m)}}(a)

as

β
−
α
→
0

\beta - \alpha \to 0

. The resulting roundoff error bounds can be several orders of magnitude less than corresponding bounds for high order differences.
A probabilistic approach to a differential-difference equation arising in analytic number theory
Authors: Chamayou Jean-Marie-François 
Published: June 2010

The differential-difference equation\[tv′(t)+v(t−1)=0,amp;t>1,v(t)=0,amp;t>0,v(t)=constant,amp;0≦t≦1,\begin {array}{*{20}{c}} {tv’(t) + v(t - 1) = 0,} \hfill & {t > 1,} \hfill \\ {v(t) = 0,} \hfill & {t > 0,} \hfill \\ {v(t) = {\operatorname {constant}},} \hfill & {0 \leqq t \leqq 1,} \hfill \\ \end {array}\]can be solved by the Monte-Carlo method, for the initial conditionv(t)=e−γ,0≦t≦1v(t) = {e^{ - \gamma }},0 \leqq t \leqq 1, where thev(t)v(t)represent the probability density of a random variable:\[t=limn→∞∑i=1n∏j=1ixj,t = \lim \limits _{n \to \infty } \sum \limits _{i = 1}^n {\prod \limits _{j = 1}^i {{x_j},} }\]where thexj{x_j}are independent and uniformly distributed on (0, 1).
Almost-interpolatory Chebyshev quadrature
Authors: Salkauskas K. 
Published: June 2010

The requirement that a Chebyshev quadrature formula have distinct real nodes is not always compatible with the requirement that the degree of precision of an n-point formula be at least equal to n. This condition may be expressed as

‖
d
‖

ν

=
0
,
1
≦
p

{\left \| d \right \|_\nu } = 0,1 \leqq p

, where

d
=
(

d
1

,
⋯
,

d
n

)

d = ({d_1}, \cdots ,{d_n})

with
\[

d
j

=

μ
0

(
ω
)

n

∑

i
=
1

n

x
i
j

−

μ
j

(
ω
)
,

j
=
1
,
2
,
⋯
,
n
,

{d_j} = \frac {{{\mu _0}(\omega )}}{n}\sum \limits _{i = 1}^n {x_i^j - {\mu _j}(\omega ),\quad j = 1,2, \cdots ,n,}

\]

μ
j

(
ω
)
,
j
=
0
,
1
,
⋯

{\mu _j}(\omega ),j = 0,1, \cdots

, are the moments of the weight function

ω
\omega

used in the quadrature, and

x
1

,
⋯
,

x
n

{x_1}, \cdots ,{x_n}

are the nodes. In those cases when

‖
d
‖

2

{\left \| d \right \|_2}

does not vanish for a real choice of nodes, it has been proposed that a real minimizer of

‖
d
‖

2

{\left \| d \right \|_2}

be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of

‖
d
‖

p

,
1
≦
p
>
∞

{\left \| d \right \|_p},1 \leqq p > \infty

, always lead to formulae that are degenerate in the sense that the nodes are not all distinct. The results are valid for a large class of weight functions.
Discrete Green’s functions
Authors: McAllister G. T. , Sabotka E. F. 
Published: June 2010

Let

G
(
P
;
Q
)

G(P;Q)

be the discrete Green’s function over a discrete h-convex region

Ω
\Omega

of the plane; i.e.,

a
(
P
)

G

x

x
¯

(
P
;
Q
)
+
c
(
P
)

G

y

y
¯

(
P
;
Q
)
=
−
δ
(
P
;
Q
)

/

h
2

a(P){G_{x\bar x}}(P;Q) + c(P){G_{y\bar y}}(P;Q) = - \delta (P;Q)/{h^2}

for

P
∈

Ω
h

,
G
(
P
;
Q
)
=
0

P \in {\Omega _h},G(P;Q) = 0

for

P
∈
∂

Ω
h

P \in \partial {\Omega _h}

. Assume that

a
(
P
)

a(P)

and

c
(
P
)

c(P)

are Hölder continuous over

Ω
\Omega

and positive. We show that

|

D

(
m
)

G
(
P
;
Q
)

|

≦

A
m

/

ρ

P

Q

m

|{D^{(m)}}G(P;Q)| \leqq {A_m}/\rho _{P\;Q}^m

and

|

D
~

(
m
)

G
(
P
;
Q
)

|

≦

B
m

d
(
Q
)

/

ρ

P

Q

m
+
1

|{\tilde D^{(m)}}G(P;Q)| \leqq {B_m}d(Q)/\rho _{P\;Q}^{m + 1}

, where

D

(
m
)

{D^{(m)}}

is an mth order difference quotient with respect to the components of P or Q, and

D
~

(
m
)

{\tilde D^{(m)}}

denotes an mth order difference quotient only with respect to the components of P.
Computation of modified Bessel functions and their ratios
Authors: Amos D. E. 
Published: June 2010

An efficient algorithm for calculating ratios

r
v

(
x
)
=

I

v
+
1

(
x
)

/

I
v

(
x
)
,
v
≧
0
,
x
≧
0

{r_v}(x) = {I_{v + 1}}(x)/{I_v}(x),v \geqq 0,x \geqq 0

, is presented. This algorithm in conjunction with the recursion relation for

r
v

(
x
)

{r_v}(x)

gives an alternative to other recursive methods for

I
v

(
x
)

{I_v}(x)

when approximations for low-order Bessel functions are available. Sharp bounds on

r
v

(
x
)

{r_v}(x)

and

I
v

(
x
)

{I_v}(x)

are also established in addition to some monotonicity properties of

r
v

(
x
)

{r_v}(x)

and

r

v

′

(
x
)

r’_{v}(x)

.
On the osculatory rational interpolation problem
Authors: Wuytack Luc 
Published: June 2010

The problem of the existence and construction of a table of osculating rational functions

r

1
,
m

{r_{1,m}}

for

1
,
m
⩾
0

1,m \geqslant 0

is considered. First, a survey is given of some results from the theory of osculatory rational interpolation of order

s
i

−
1

{s_i} - 1

at points

x
i

{x_i}

for

i
⩾
0

i \geqslant 0

. Using these results, we prove the existence of continued fractions of the form
\[

c
0

+

c
1

⋅
(
x
−

y
0

)
+
…
+

c
k

⋅
(
x
−

y
0

)
…
(
x
−

y

k
−
1

)
+

c

k
+
1

⋅
(
x
−

y
0

)
…
(
x
−

y
k

)

1

+

c

k
+
2

⋅
(
x
−

y

k
+
1

)

1

+

c

k
+
3

⋅
(
x
−

y

k
+
2

)

1

+
…
,

{c_0} + {c_1} \cdot (x - {y_0}) + \ldots + {c_k} \cdot (x - {y_0}) \ldots (x - {y_{k - 1}}) + \frac {{{c_{k + 1}} \cdot (x - {y_0}) \ldots (x - {y_k})}}{1} + \frac {{{c_{k + 2}} \cdot (x - {y_{k + 1}})}}{1} + \frac {{{c_{k + 3}} \cdot (x - {y_{k + 2}})}}{1} + \ldots ,

\]
with the

y
k

{y_k}

suitably selected from among the

x
i

{x_i}

, whose convergents form the elements

r

k
,
0

,

r

k
+
1
,
0

,

r

k
+
1
,
1

,

r

k
+
2
,
1

,
…

{r_{k,0}},{r_{k + 1,0}},{r_{k + 1,1}},{r_{k + 2,1}}, \ldots

of the table. The properties of these continued fractions make it possible to derive an algorithm for constructing their coefficients

c
i

{c_i}

for

i
⩾
0

i \geqslant 0

. This algorithm is a generalization of the qd-algorithm.
Some properties of orthogonal polynomials
Authors: Hunter D. B. 
Published: June 2010

Some results are obtained concerning the signs of the coefficients in the expansions in powers of

x

−
1

,

(
1
+
x

)

−
1

{x^{ - 1}},{(1 + x)^{ - 1}}

or

(
1
−
x

)

−
1

{(1 - x)^{ - 1}}

of

1

/

p
n

(
x
)

1/{p_n}(x)

and

q
n

(
x
)

{q_n}(x)

, where

p
n

(
x
)

{p_n}(x)

is the polynomial of degree n in the orthogonal sequence associated with a given weight-function

w
(
x
)

w(x)

over

(
−
1
,
1
)

( - 1,1)

and

q
n

(
x
)
=

∫

−
1

1

w
(
t
)

p
n

(
t
)

(
x
−
t

)

−
1

d
t

{q_n}(x) = \smallint _{ - 1}^1w(t){p_n}(t){(x - t)^{ - 1}}dt

.
A necessary and sufficient condition for transcendency
Authors: Mahler K. 
Published: June 2010

As has been known for many years (see, e.g., K. Mahler, J. Reine Angew. Math., v. 166, 1932, pp. 118-150), a real or complex number

ζ
\zeta

is transcendental if and only if the following condition is satisfied. To every positive number

ω
\omega

there exists a positive integer n and an infinite sequence of distinct polynomials

{

p
r

(
z
)
}
=
{

p

r
0

+

p

r
1

z
+
⋯
+

p

r
n

z
n

}

\{ {p_r}(z)\} = \{ {p_{{r_0}}} + {p_{{r_1}}}z + \cdots + {p_{{r_n}}}{z^n}\}

at most of degree n with integral coefficients, such that
\[

0
>

|

p
r

(
ζ
)

|

⩽

{

p

r
0

2

+

p

r
1

2

+
⋯
+

p

r
n

2

}

−
ω

f
o
r

a
l
l

r
.

0 > |{p_r}(\zeta )| \leqslant {\{ p_{{r_0}}^2 + p_{{r_1}}^2 + \cdots + p_{{r_n}}^2\} ^{ - \omega }}\quad for\;all\quad r.

\]
In the present note I prove a simpler test which makes the transcendency of

ζ
\zeta

depend on the approximation behaviour of a single sequence of distinct polynomials of arbitrary degrees with integral coefficients.