On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth

Devendra Kumar; Azza M. Alghamdi

doi:10.1515/dema-2024-0096

Article Open Access

On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth

Devendra Kumar and Azza M. Alghamdi

Published/Copyright: February 18, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

It has been noted that lower type of an entire function completely ignores the value of lower order. The question arises for entire functions of irregular growth that what happens when we replace order by an arbitrary nonzero finite number. Here in this article our aim is to solve this problem by defining new lower ( p , q ) -type by using a ( p , q ) -scale, ( p ≥ q ≥ 1 ) for an entire function. Moreover, a relationship has been established between lower ( p , q ) -type of entire function solutions of linear homogeneous partial differential equation of second order with coefficients occurring in series expansion and Laguerre polynomial approximation errors in sup norm.

Keywords: irregular growth; Laguerre polynomial approximation errors; homogeneous partial differential equations; entire functions and finite bi-disk

MSC 2010: 30E10; 41A10

1 Introduction

The behavior of meromorphic solutions of the following homogeneous linear partial differential equations of the second order:

(1.1) t 2 ∂ 2 u ∂ t 2 − z 2 ∂ 2 u ∂ z 2 + ( 2 t + 2 ) ∂ u ∂ t − 2 z ∂ u ∂ z = 0 ,

(1.2) t 2 ∂ 2 u ∂ t 2 − z 2 ∂ 2 u ∂ z 2 + t ∂ u ∂ t − z ∂ u ∂ z + t 2 u = 0

was studied by Hu and Yang [1,2]. They investigated that these solutions are closely related to Bessel functions and Bessel polynomials for ( t , z ) ∈ C 2 . Wang et al. [3] studied the growth parameters-order and the type of entire function solutions of the partial differential equation

(1.3) t ∂ 2 u ∂ t 2 + ( δ + 1 − t ) ∂ u ∂ t + z ∂ u ∂ z = 0

for real δ > 0 . These solutions are related to Laguerre polynomials. Wang et al. [3] proved that the PDE (1.3) has an entire solution u = f ( t , z ) on C 2 , if and only if u = f ( t , z ) has a series expansion f ( t , z ) = ∑ n = 0 ∞ a n L n ( δ ) ( t ) z n .

Here,

L n ( δ ) ( t ) = ∑ k = 0 n n + δ n − k ( − t ) k k !

are the Laguerre polynomials. Laguerre polynomials L n ( δ ) ( t ) are well-known in the Gaussian quadrature to numerically compute integrals of the form

∫ 0 ∞ t δ e − t f ( t ) d t ( δ > − 1 ) .

Laguerre expansions have many uses in the Mathieu equation, prolate spheroidal wave equation, Laplace’s tidal equation, quantum mechanics, Vlasov-Maxwell’s equation, etc.

The function f ( t ) can be expanded in the form of a series of { L n ( δ ) ( t ) } 0 ∞ [1,4–8]

f ( t ) = ∑ n = 0 ∞ a n L n ( δ ) ( t ) , a n = n ! Γ ( n + δ + 1 ) ∫ 0 ∞ t δ e − t L n ( δ ) ( t ) f ( t ) d t .

The decay of the coefficients of f ( t ) expanded in an orthogonal polynomial series in a finite interval has been extensively studied (see [4,5,9,10]). Unlike most other sets of orthogonal polynomials in a finite interval, the Laguerre polynomials increase exponentially with the degree n , so it is difficult to work with unnormalized functions without encountering overflow [4,11].

The Bernstein theorem identifies a real analytic function on the closed unit disk as the restriction of an analytic function defined on an open disk of radius R > 1 by computing R from the sequence of minimal errors generated from optimal polynomial approximates. The disk of the maximum radius on which the analytic function f ( t , z ) exists is denoted by ( D R × D R ) .

In a neighborhood of origin, the function f ( t , z ) has the local expansion

(1.4) f ( t , z ) = ∑ n = 0 ∞ w n ( t ) n ! z n ,

where w n ( t ) = ∂ n f ∂ z n ( t , 0 ) is an entire solution of the ordinary differential equation

(1.5) t d 2 w d t 2 + ( δ + 1 − t ) d w d t + n w = 0 ,

where w n ( t ) z n = n ! a n L n ( δ ) ( t ) z n . Following the method of Wang and Guo [12], a second independent solution of (1.5) can be obtained as

(1.6) X n ( δ ) ( t ) = q L n ( δ ) ( t ) log t + ∑ i = 0 ∞ p i t i ,

where q ≠ 0 and p i are constants. So, there exist a n and b n satisfying

(1.7) w n ( t ) = n ! a n L n ( δ ) ( t ) + b n X n ( δ ) ( t ) .

Because of the singularity of X n ( δ ) ( t ) at t = 0 , it implies to b n = 0 . Hence,

(1.8) f ( t , z ) = ∑ n = 0 ∞ a n L n ( δ ) ( t ) z n .

A function f ( t , z ) is said to be regular in ( D R × D R ) if the series (1.8) converges uniformly on compact subsets of ( D R × D R ) . A class of functions f ( t , z ) regular in ( D R × D R ) will be denoted by A ( D R × D R ) . If f is an entire function, it has no singularities in the finite positive C 2 plane and write f ∈ A ( C 2 ) .

Let Π n be a set of Laguerre polynomials of degree no higher than n . Approximation of function f ( t , z ) ∈ A ( D R × D R ) by Laguerre polynomials L n ( δ ) ( t , z ) ∈ Π n can be determined as

(1.9) E n ( f , R ) = inf L n ( δ ) ( t , z ) ∈ Π n { max ( t , z ) ∈ D R ¯ ∣ f ( t , z ) − L n ( δ ) ( t , z ) ∣ } ,

where D R ¯ is the closure of D R .

It has been shown [3] that the partial differential equation (1.5) has an entire solution w = f ( t , z ) on C 2 , if and only if w = f ( t , z ) has a series expansion (1.8) such that

limsup n → ∞ ∣ a n ∣ 1 n = 0 .

The concept of order and type was generalized in the literature [13,14] by introducing ( p , q ) -scale, p ≥ q ≥ 1 . Following the concept of index-pair ( p , q ) introduced by Juneja et al. [13,14], we have the following definitions for an entire function f ( t , z ) = ∑ n = 0 ∞ a n L n δ ( t ) z n .

Definition 1.1

An entire function f ( t , z ) is said to be of ( p , q ) -order ρ ( p , q ) and lower ( p , q ) -order λ ( p , q ) if it is of index-pair ( p , q ) such that

ρ ( p , q ) = limsup r → ∞ log [ p ] M ( r , r , f ) log [ q ] r ; λ ( p , q ) = liminf r → ∞ log [ p ] M ( r , r , f ) log [ q ] r ,

where M ( r , r , f ) ≔ sup t , z ∈ D R ¯ ∣ f ( t , z ) ∣ , r < R .

The function f ( t , z ) of ( p , q ) -order ρ ( p , q ) ( b < ρ ( p , q ) < ∞ ) is said to be of ( p , q ) -type T ( p , q ) and lower ( p , q ) -type t ( p , q ) if

(1.10) T ( p , q ) = limsup r → ∞ log [ p − 1 ] M ( r , r , f ) ( log [ q − 1 ] r ) ρ ( p , q ) ; t ( p , q ) = liminf r → ∞ log [ p − 1 ] M ( r , r , f ) ( log [ q − 1 ] r ) ρ ( p , q ) ,

where b = 1 if p = q and b = 0 if p > q .

It is known that an entire function is of irregular growth if ρ ( p , q ) ≠ λ ( p , q ) . In equation (1.10), the lower ( p , q ) -type t ( p , q ) completely ignores the value of lower ( p , q ) -order λ ( p , q ) . Now, the question arises that what happens when we replace ρ ( p , q ) by an arbitrary number γ ( p , q ) ( b < γ ( p , q ) < ∞ ) in equation (1.10) for lower ( p , q ) -type t ( p , q ) . We solve this problem by defining the lower ( p , q ) -type of f ( t , z ) depending on lower ( p , q ) -order λ ( p , q ) by

t λ ( p , q ) = liminf r → ∞ log [ p − 1 ] M ( r , r , f ) ( log [ q − 1 ] r ) λ ( p , q ) , b < λ ( p , q ) < ∞ .

McCoy [15,16] studied classical order and type of entire function solutions of certain second-order linear partial differential equations in terms of approximation errors in sup norm. These solutions are known as GBSPs (generalized bi-axially symmetric potentials). Kasana and Kumar [17] studied the growth of entire function GBSP in terms of approximation error in L p -norm ( 1 ≤ p < ∞ ) on the Carathéodory domain. In [18], Kumar considered entire function solutions of the Helmholtz equation in R 2 and obtained some lower bounds on classical order and type. In a subsequent paper [19], Kumar obtained the characterization of growth parameters in terms of axially symmetric harmonic polynomial and Lagrange polynomial approximation errors in n -dimensions. Srivastava [20] investigated the order and type of entire function GBSP in terms of series expansion coefficients and approximation errors. For GBSP’s functions, there is a large literature concerning the growth and approximation error in different norms. All these results are related to Jacobi and Gegenbauer polynomials. But, there are a few studies concerning the growth and approximation related to Laguerre polynomials. Therefore, in this article, our aim is to establish relationship between the lower type t γ ( p , q ) with coefficients occurring in series expansion (1.8) and Laguerre polynomial approximation errors defined by (1.9) in sup norm.

2 Auxiliary results

In this section we will prove some lemmas that will be used in the sequel.

Lemma 2.1

Let f ( t , z ) ∈ A ( D R × D R ) , then the following inequality holds:

∣ a n ∣ R n ≤ 2 n ! Γ ( n + δ + 1 ) 1 2 E n − 1 ( f , R ) ,

where E n − 1 ( f , R ) is determined by (1.9).

Proof

Using the orthogonality property of Laguerre polynomials [10] with uniform convergence of the series (1.8), we have

(2.1) a n R n = n ! Γ ( n + δ + 1 ) ∫ 0 ∞ e − t t δ f ( t , z ) L n ( δ ) ( t ) d t .

On the basis of the addition theorem of Laguerre polynomials L n ( δ ) ( t ) , we obtain

(2.2) ∫ 0 ∞ g ( τ , z ) e − t t δ L n ( δ ) ( t ) d t = 0 ,

where g ∈ Π n − 1 , 0 < τ < R . In view of (2.2), we can rewrite (2.1) as

(2.3) a n τ n = n ! Γ ( n + δ + 1 ) ∫ 0 ∞ e − t t δ ( f ( t , z ) − g ( τ , z ) ) L n ( δ ) ( t ) d t .

Using the Schwartz inequality and orthogonality of Laguerre polynomials in (2.3), we obtain

(2.4) a n τ n ≤ max τ , z ∈ D R ¯ ∣ f ( t , z ) − g ( τ , z ) ∣ n ! Γ ( n + δ + 1 ) 1 2 .

Now, it follows from the definition of E n ( H , R ) that there exists a Laguerre polynomial g ˜ ∈ Π n − 1 , for which

(2.5) max τ , z ∈ D R ¯ ∣ f ( t , z ) − g ˜ ( τ , z ) ∣ ≤ 2 E n − 1 ( f , R ) .

Substituting g = g ˜ in (2.4) and considering inequality (2.5) as well as arbitrariness of τ , we obtain the lemma from (2.4).□

Lemma 2.2

For an entire function f ( t , z ) ∈ A ( C 2 ) , the following inequality holds:

E n ( f , R ) ≤ K M ( r , r , f ) ( n + 2 ) δ 2 R r n

for all r > e R and all sufficiently large values of n. Here, K is a constant independent of n and r.

Proof

Let us consider the truncated polynomial

P n f ( t ) = ∑ k = 0 n a k L k ( δ ) ( t ) z k ,

where r > 0 and P n f ∈ Π n . Considering the approximation error E n ( f , R ) determined by (1.9) for all r , 0 < r < R , we obtain

(2.6) E n ( f , R ) ≤ max t , z ∈ D R ¯ ∣ f ( t , z ) − P n f ( t , z ) ∣ ≤ ∑ j = n + 1 ∞ ∣ a j ∣ R j ∣ L j ( δ ) ( t ) ∣ .

We have

(2.7) ∣ a k ∣ ≤ M ( r , r , f ) r k k ! Γ ( k + δ + 1 ) 1 2

for every r < R , and from [11, p. 31], [21, p. 786]

(2.8) ∣ e − t 2 L n ( δ ) ( t ) ∣ ≤ Γ ( 1 + δ + n ) n ! Γ ( 1 + δ ) , δ > 0 , t ≥ 0 , n = 0 , 1 , … .

Now, combining (2.6), (2.7), and (2.8), we obtain

(2.9) E n ( f , R ) ≤ M ( r , r , f ) e t 2 ∑ j = n + 1 ∞ j ! Γ ( j + δ + 1 ) 1 2 Γ ( j + δ + 1 ) j ! Γ ( δ + 1 ) R r j .

Since Γ ( x + a ) Γ ( x ) ∼ x a as x → ∞ . Now, from (2.9), we obtain

E n ( f , R ) ≤ M ( r , r , f ) Γ ( δ + 1 ) e t 2 ∑ j = n + 1 ∞ ( j + 1 ) δ 2 R r j = M ( r , r , f ) Γ ( δ + 1 ) R r n e t 2 ∑ j = n + 1 ∞ ( j + 1 ) δ 2 R r ( j − n ) ≤ M ( r , r , f ) Γ ( δ + 1 ) R r n e t 2 e n ( n + 2 ) δ 2 e − ( n + 1 ) ∑ j = 0 ∞ 1 + j j 0 δ 2 ( e ) ( − j ) , n > j 0 = K M ( r , r , f ) ( n + 2 ) δ 2 R r n .

Hence, the proof is completed.□

3 Main results

In this section, we will prove our main results.

Theorem 3.1

Let f ( t , z ) = ∑ n = 0 ∞ a n L n δ ( t ) z n be an entire function and γ ( p , q ) ( b < γ ( p , q ) < ∞ ) an arbitrary number such that

t γ ( p , q ) = liminf r → ∞ log [ p − 1 ] M ( r , r , f ) ( log [ q − 1 ] r ) γ ( p , q ) .

Then

(3.1) t γ ( p , q ) M ≥ liminf n → ∞ log [ p − 2 ] n ( log [ q − 1 ] ∣ a n ∣ − 1 n ) γ ( p , q ) − A .

M = 1 e ρ i f ( p , q ) = ( 2 , 1 ) , ( γ − 1 ) γ − 1 γ γ i f ( p , q ) = ( 2 , 2 ) , 1 f o r ( p , q ) ≠ ( 2 , 1 ) a n d ( 2 , 2 ) .

A = 1 if ( p , q ) = ( 2 , 2 ) and A = 0 for all other index-pair ( p , q ) . Further, if ∣ a n a n + 1 ∣ forms a non-decreasing function of n for n > n 0 , then

(3.2) t γ ( p , q ) M = liminf n → ∞ log [ p − 2 ] n ( log [ q − 1 ] ∣ a n ∣ − 1 n ) γ ( p , q ) − A .

Proof

Since a n L n δ a n + 1 L n + 1 δ ≈ a n a n + 1 forms a non-decreasing function of n for n > n 0 , the proof can be easily obtained on the same lines by putting λ n = n , as given by Juneja et al. [14] for lower ( p , q ) -type of an entire function in terms of coefficients occurring in its gap power series expansion, i.e., for γ ≡ γ ( p , q ) = ρ ( p , q ) .□

Theorem 3.2

Let f ( t , z ) = ∑ n = 0 ∞ a n L n δ ( t ) z n be an entire function of ( p , q ) -order ρ ( p , q ) , and γ ( p , q ) ( b < γ ( p , q ) < ∞ ) be an arbitrary number, then

(3.3) t γ ( p , q ) ≥ liminf n → ∞ log [ p − 2 ] n γ ( p , q ) ( log [ q − 2 ] ∣ a n a n + 1 ∣ ) γ ( p , q ) − A .

(3.4) liminf n → ∞ log [ p − 2 ] n ρ ( p , q ) ( log [ q − 2 ] ∣ a n a n + 1 ∣ ) ρ ( p , q ) − A = 0 , i f ρ ( p , q ) ≠ λ ( p , q ) ,

where λ ( p , q ) is the lower ( p , q ) -order of f ( t , z ) .

Proof

Let

S = liminf n → ∞ log [ p − 2 ] n γ ( p , q ) ( log [ q − 2 ] ∣ a n a n + 1 ∣ ) γ ( p , q ) − A .

By the definition S ≥ 0 . If S = 0 , then inequality (3.3) holds. Hence, let S > 0 in this case for given ε > 0 and n > n o , we have

exp [ q − 1 ] log [ p − 1 ] ( n ) − log ( S − ε ) γ ( p , q ) − A > log a n a n + 1 .

Putting n = m o , m o + 1 , … , m − 1 and adding the inequalities thus obtained, we obtain

log ∣ a m o a m ∣ < ∑ n = m o m − 1 exp [ q − 1 ] log [ p − 1 ] ( n ) − log ( S − ε ) γ ( p , q ) − A = ∑ n = m o m − 1 F ( n ) = ( m − m o ) F ( m − 1 ) − ∑ n = m o + 1 m − 1 F ( n ) ,

where F ( t ) = exp [ q − 1 ] log [ p − 1 ] ( t ) − log ( S − ε ) γ ( p , q ) − A and n ( t ) = n if m − 2 < t ≤ m − 1 . Hence,

log a m o a m < − ( m − m o ) F ( m − 1 ) − ∫ m 0 m − 1 t d ( F ( t ) ) .

For ( p , q ) = ( 2 , 1 ) , we have

log a m a m o > − ( m − m o ) F ( m − 1 ) + ∫ m o m − 1 t d log ( S − ε ) − log t γ ( 2 , 1 ) , log ∣ a m ∣ > − ( m − m o ) F ( m − 1 ) + ∫ m o m − 1 d t γ ( 2 , 1 ) > − ( m − m o ) F ( m − 1 ) − ( m − 1 − m o ) γ ( 2 , 1 ) ,

log ∣ a m ∣ − 1 m < O ( 1 ) + F ( m − 1 ) − 1 γ ( 2 , 1 ) + ε ≤ O ( 1 ) + 1 γ ( 2 , 1 ) log m − 1 S − ε − 1 γ ( 2 , 1 ) + ε .

Hence

(3.5) ( S − ε ) ≤ ( 1 + O ( 1 ) ) m − 1 [ ∣ a m ∣ − 1 m ] γ ( 2 , 1 ) ≤ t ( 2 , 1 ) γ ( 2 , 1 ) .

For ( p , q ) = ( 2 , 2 ) , inequality (3.3) yields

log a m a m o > O ( 1 ) − m F ( m − 1 ) + ∫ m o + 1 m − 1 t d [ F ( t ) ] > O ( 1 ) − m F ( m − 1 ) + ( m − 1 ) F ( m − 1 ) + ∫ m o + 1 m − 1 F ( t ) d t ≈ O ( 1 ) − m F ( m − 1 ) + ( m − 1 ) F ( m − 1 ) − ( γ ( 2 , 2 ) − 1 ) γ ( 2 , 2 ) ( S − ε ) 1 γ ( 2 , 2 ) − 1 [ t ] m o + 1 m − 1

(3.6) log ∣ a m ∣ − 1 m < 1 m m − 1 S − ε 1 γ ( 2 , 2 ) − 1 + ( 1 + O ( 1 ) ) ( γ ( 2 , 2 ) − 1 ) ( m − 1 ) γ ( 2 , 2 ) ( γ ( 2 , 2 ) − 1 ) γ ( 2 , 2 ) m ( S − ε ) 1 ( γ ( 2 , 2 ) − 1 ) ≈ γ ( 2 , 2 ) − 1 γ ( 2 , 2 ) m − 1 S − ε 1 γ ( 2 , 2 ) − 1 , S − ε < ( 1 + O ( 1 ) ) γ ( 2 , 2 ) − 1 γ ( 2 , 2 ) γ ( 2 , 2 ) − 1 m − 1 [ log ∣ a m ∣ − 1 m ] γ ( 2 , 2 ) − 1 ≤ t γ ( 2 , 2 ) γ ( 2 , 2 ) .

Finally, consider the case when ( p , q ) ≠ ( 2 , 1 ) and ( 2 , 2 ) . In this situation, (3.4) reduced to

log a m − 1 a m < O ( 1 ) + m F ( m − 1 ) + ∫ m o + 1 m − 1 t d ( F ( t ) ) < O ( 1 ) + m F ( m − 1 ) − ( m − 1 ) F ( m − 1 ) < O ( 1 ) + m F ( m − 1 ) − ( m o + 1 ) F ( m − 1 )

(3.7) log ∣ a m ∣ − 1 m > ( 1 + O ( 1 ) ) F ( m − 1 ) = ( 1 + O ( 1 ) ) exp [ q − 1 ] log [ p − 1 ] ( m − 1 ) − log ( S − ε ) γ ( p , q ) − A .

Proceeding to limits in (3.5), (3.6), and (3.7), we obtain (3.3) which is true when S = 0 ; if S = ∞ , then t γ ( p , q ) = ∞ . For γ ( p , q ) > λ ( p , q ) , t γ ( p , q ) = 0 and (3.4) follows since ρ ( p , q ) > λ ( p , q ) .□

Theorem 3.3

The function f ( t , z ) ∈ A ( D R × D R ) continue to an entire function f ( t , z ) if and only if the following equality holds:

(3.8) lim n → ∞ [ E n ( f , R ) ] 1 n = 0 .

Proof

Let f ( t , z ) ∈ A ( D R × D R ) continues to an entire function f ( t , z ) . Then, equality (3.8) follows from Lemma 2.2. For the proof of only if part, we use the estimate (2.6) with Lemma 2.1, we obtain

(3.9) ∑ n = 0 ∞ a n L n ( δ ) ( t ) z n ≤ ∣ a 0 ∣ + e t 2 ∑ n = 1 ∞ ∣ a n ∣ r n Γ ( n + δ + 1 ) Γ ( n + 1 ) Γ ( δ + 1 ) ≤ ∣ a 0 ∣ + 2 e t 2 Γ ( δ + 1 ) ∑ n = 1 ∞ Γ ( n + 1 ) Γ ( n + δ + 1 ) 1 2 Γ ( n + δ + 1 ) Γ ( n + 1 ) E n − 1 ( f , R ) r R n = ∣ a 0 ∣ + 2 e t 2 Γ ( δ + 1 ) ∑ n = 1 ∞ Γ ( n + δ + 1 ) Γ ( n + 1 ) 1 2 E n − 1 ( f , R ) r R n = ∣ a 0 ∣ + 2 e t 2 Γ ( δ + 1 ) ∑ n = 1 ∞ ( n + 1 ) δ 2 E n − 1 ( f , R ) r R n ,

hence by (3.8), a uniform convergence of the series in the right side of equality (1.8) on compact subsets of the complex plane follows. Hence, the function f ( t , z ) ∈ A ( D R × D R ) represented by a series (1.8) shall continue over the whole complex plane C .□

Theorem 3.4

Let f ( t , z ) ∈ A ( D R × D R ) continue to an entire function f ( t , z ) ∈ A ( C 2 ) and γ ( p , q ) ( b < γ ( p , q ) < ∞ ) an arbitrary number such that (1.10) satisfied. Then,

(3.10) t γ ( p , q ) M ≥ liminf n → ∞ log [ p − 2 ] n ( log [ q − 1 ] [ R − n E n ( f , R ) ] − 1 n ) γ ( p , q ) − A .

Further, if E n ( f , R ) E n + 1 ( f , R ) forms a non-decreasing function of n for n > n 0 , then

(3.11) t γ ( p , q ) M = liminf n → ∞ log [ p − 2 ] n ( log [ q − 1 ] [ R − n E n ( f , R ) ] − 1 n ) γ ( p , q ) − A .

Proof

To prove Theorem 3.4, we first consider the entire functions of complex variable z,

f 1 ( t , z ) = ∑ n = 0 ∞ 1 K ( n + 2 ) δ 2 E n ( f , R ) z R n , f 2 ( t , z ) = ∑ n = 1 ∞ 2 ( n + 1 ) − δ 2 E n − 1 ( f , R ) z R n .

For r > e R , by Lemma 2.2 and inequality (3.9), we obtain

(3.12) m ( r , r , f 1 ) ≤ M ( r , r , f ) ≤ ∣ a 0 ∣ + M ( r , r , f 2 ) ,

where m ( r , r , f 1 ) is the maximum term of a power series of the entire function f 1 ( t , z ) on bi-disk D R × D R , and M ( r , r , f 2 ) ≔ max z , t ∈ D R ¯ ∣ f 2 ( z , t ) ∣ . Hence, by using (3.12), we obtain

(3.13) log m ( r , r , f 1 ) ≤ log M ( r , r , f ) ≤ log M ( r , r , f 2 ) .

Using the result of Valiron [22] on the maximum term m ( r , r , f 1 ) , we obtain

log M ( r , r , f 1 ) ≃ log m ( r , r , f 1 ) as r → ∞ .

Hence,

(3.14) t γ ( p , q , f 1 ) ≤ t γ ( p , q , f ) ≤ t γ ( p , q , f 2 ) .

Now, using formulas (3.1) and (3.2) for ( p , q ) -type t γ ( p , q ) of an entire function f ( t , z ) , we obtain the following inequality:

(3.15) t γ ( p , q , f 1 ) M = t γ ( p , q , f 2 ) M ≥ liminf n → ∞ log [ p − 2 ] n ( log [ q − 1 ] [ R − n E n ( f , R ) ] − 1 n ) γ ( p , q ) − A

and

(3.16) t γ ( p , q , f 1 ) M = t γ ( p , q , f 2 ) M = liminf n → ∞ log [ p − 2 ] n ( log [ q − 1 ] [ R − n E n ( f , R ) ] − 1 n ) γ ( p , q ) − A .

On combining (3.13) with (3.14) and (3.15), we complete the required proof.□

Theorem 3.5

Let f ( t , z ) ∈ A ( D R × D R ) continue to an entire function f ( t , z ) ∈ A ( C 2 ) having ( p , q ) -order ρ ( p , q ) and γ ( p , q ) ( b < γ ( p , q ) < ∞ ) an arbitrary number, then

(3.17) t γ ( p , q ) ≥ liminf n → ∞ log [ p − 2 ] n γ ( p , q ) ( log [ q − 2 ] [ E n ( f , R ) E n + 1 ( f , R ) ] ) γ ( p , q ) − A .

(3.18) liminf n → ∞ log [ p − 2 ] n ρ ( p , q ) ( log [ q − 2 ] [ E n ( f , R ) E n + 1 ( f , R ) ] ) ρ ( p , q ) − A = 0 , i f ρ ( p , q ) ≠ λ ( p , q ) ,

where λ ( p , q ) is the lower ( p , q ) -order of f ( t , z ) .

Proof

The proof can be obtained by using the same reasoning as in Theorem 3.2 with inequalities (3.14), (3.3), and (3.4).□

dsingh@bu.edu.sa, amhalghamdi@hotmail.com

Acknowledgements

The authors are grateful for the reviewers for giving valuable comments to improve the manuscript.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animals use.
Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on responsible request.

References

[1] P. C. Hu and C. C. Yang, Global solutions of homogeneous linear partial differential equations of the second order, Michigan Math. J. 58, (2009), no. 3, 807–831, DOI: https://doi.org/10.1307/mmj/1260475702. 10.1307/mmj/1260475702Search in Google Scholar

[2] P. C. Hu and C. C. Yang, A linear homogeneous partial differential equation with entire solutions represented by Bessel polynomials, J. Math. Anal. Appl. 368 (2010), no. 1, 263–280, DOI: https://doi.org/10.1016/j.jmaa.2010.03.048. 10.1016/j.jmaa.2010.03.048Search in Google Scholar

[3] X. Li Wang, F. Li Zhang, P. Chu Hu, A linear homogeneous partial differential equation with entire solutions represented by Laguerre polynomials, Abstr. Appl. Anal. 2012 (2012), 1–10, DOI: https://doi.org/10.1155/2012/609862. 10.1155/2012/609862Search in Google Scholar

[4] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, New York, 2000. Search in Google Scholar

[5] G. Dahlquist and A. Björck, Numerical Methods in Scientific Computing, SIAM, Philadelphia, 2007. 10.1137/1.9780898717785Search in Google Scholar

[6] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second ed., Academic Press, New York, 1984. 10.1016/B978-0-12-206360-2.50012-1Search in Google Scholar

[7] W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983), 1170–1186, DOI: https://doi.org/10.1137/0720087. 10.1137/0720087Search in Google Scholar

[8] I. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000. 10.1137/1.9780898719598Search in Google Scholar

[9] S. Xiang, X. Chen, and H. Wang, Error bounds for approximation in Chebyshev points, Numer. Math. 116 (2010), 463–491, DOI: https://doi.org/10.1007/s00211-010-0309-4. 10.1007/s00211-010-0309-4Search in Google Scholar

[10] S. Xiang, On error bounds for orthogonal polynomial expansions and Gauss-type quadrature, Technic Report, Central South University, 2010. Search in Google Scholar

[11] A. Glaser, X. Liu, and V. Rokhlin, A fast algorithm for the calculation of the roots of special functions, SIAM J. Sci. Comput. 29 (2007), 1420–1438, DOI: https://doi.org/10.1137/06067016X. 10.1137/06067016XSearch in Google Scholar

[12] Z. X. Wang and D. R. Guo, Introduction to Special Function, Peking University Press, Beijing, China, 2000. Search in Google Scholar

[13] O. P. Juneja, G. P. Kapoor, S. K. Bajpai, On the (p,q)-order and lower (p,q)-order of an entire function, J. Reine. Angew. Math. 282 (1976), 53–67, DOI: https://doi.org/10.1515/crll.1976.282.53. 10.1515/crll.1976.282.53Search in Google Scholar

[14] O. P. Juneja, G. P. Kapoor, and S. K. Bajpai, On the (p,q) -type and lower (p,q)-type of an entire function, J. Reine. Angew. Math. 290 (1977), 180–190. 10.1515/crll.1977.290.180Search in Google Scholar

[15] P. A. McCoy, Polynomial approximation of generalized biaxisymmetric potentials, J. Approx. Theory 25 (1979), no. 2, 153–168, DOI: https://doi.org/10.1016/0021-9045(79)90005-4. 10.1016/0021-9045(79)90005-4Search in Google Scholar

[16] P. A. McCoy, Optimal approximation and growth of solutions to a class of elliptic partial differential differential equations, J. Math. Anal. Appl. 154 (1991), no. 1, 203–211, DOI: https://doi.org/10.1016/0022-247X(91)90080-J. 10.1016/0022-247X(91)90080-JSearch in Google Scholar

[17] H. S. Kasana and D. Kumar, Lp-approximation of generalized bi-axially symmetric potentials over Carathéodory domains, Math. Slovaca 55 (2005), no. 5, 563–572, http://dml.cz/dmlcz/133146. Search in Google Scholar

[18] D. Kumar, On the (p,q)-growth of entire function solutions of Helmholtz equation, J. Nonlinear Sci. Appl. 4 (2011), no. 2, 92–101, DOI: http://dx.doi.org/10.22436/jnsa.004.02.01. 10.22436/jnsa.004.02.01Search in Google Scholar

[19] D. Kumar, Growth and approximation of solutions to a class of certain linear partial differential equations in RN, Math. Slovaca 64 (2014), no. 1, 139–154, DOI: https://doi.org/10.2478/s12175-013-0192-4. 10.2478/s12175-013-0192-4Search in Google Scholar

[20] G. S. Srivastava, On the growth and polynomial approximation of generalized biaxisymmetric potentials, Soochow J. Math. 23 (1997), no. 4, 347–358. Search in Google Scholar

[21] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. Search in Google Scholar

[22] G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea Publ. Co., New York, 1949. Search in Google Scholar

Received: 2023-01-05

Revised: 2024-09-30

Accepted: 2024-10-15

Published Online: 2025-02-18

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2024-0096

Keywords for this article

irregular growth; Laguerre polynomial approximation errors; homogeneous partial differential equations; entire functions and finite bi-disk

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