Skip to main content
MathOverflow

Return to Question

removed capitals from title, added tag
Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Relation between the local maxima and the local minima for approximating the Generalizedgeneralized Laguerre polynomial

added 1 character in body
Source Link
Glorfindel
Glorfindel
  • 2.8k
  • 6
  • 28
  • 38

I have already asked my question in the link below: https://mathoverflow.net/questions/418497/minima-approximation-for-laguerre-polynomials?noredirect=1#comment1074404_418497 I have suggested to anyone to give me the approximations of the minima for the laguerreLaguerre polynomial, but someone have suggested the zero Laguerre polynomials. But i want the general formula for the approximation of the local minima for the generalized Laguerre polynomial. Since, we have the formula given in the Book "Orthogonal polynomials-Szego 1939" in page 240: $$Max_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{\alpha/2-1/12} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ My question is: Can the local minima of the Laguerre polynomial be written as the following $$Min_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{k} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ where $-1 \leq k \leq \alpha/2-1/12$? If so why i? I have seen that because for that case $n^{k} \leq n^{\alpha/2}$. But iI want someone to suggest to me a demonstration if that approximation is true? If not can we deduce another approximation?

I have already asked my question in the link below: https://mathoverflow.net/questions/418497/minima-approximation-for-laguerre-polynomials?noredirect=1#comment1074404_418497 I have suggested to anyone to give me the approximations of the minima for the laguerre polynomial, but someone have suggested the zero Laguerre polynomials. But i want the general formula for the approximation of the local minima for the generalized Laguerre polynomial. Since, we have the formula given in the Book "Orthogonal polynomials-Szego 1939" in page 240: $$Max_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{\alpha/2-1/12} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ My question is: Can the local minima of the Laguerre polynomial be written as the following $$Min_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{k} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ where $-1 \leq k \leq \alpha/2-1/12$? If so why i have seen that because for that case $n^{k} \leq n^{\alpha/2}$. But i want someone to suggest to me a demonstration if that approximation is true? If not can we deduce another approximation?

I have already asked my question in the link below: https://mathoverflow.net/questions/418497/minima-approximation-for-laguerre-polynomials?noredirect=1#comment1074404_418497 I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, but someone have suggested the zero Laguerre polynomials. But i want the general formula for the approximation of the local minima for the generalized Laguerre polynomial. Since, we have the formula given in the Book "Orthogonal polynomials-Szego 1939" in page 240: $$Max_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{\alpha/2-1/12} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ My question is: Can the local minima of the Laguerre polynomial be written as the following $$Min_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{k} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ where $-1 \leq k \leq \alpha/2-1/12$? If so why? I have seen that because for that case $n^{k} \leq n^{\alpha/2}$. But I want someone to suggest to me a demonstration if that approximation is true? If not can we deduce another approximation?

Source Link
Assinisa Hamidata
Assinisa Hamidata
  • 99
  • 4

Relation between the local maxima and the local minima for approximating the Generalized Laguerre polynomial

I have already asked my question in the link below: https://mathoverflow.net/questions/418497/minima-approximation-for-laguerre-polynomials?noredirect=1#comment1074404_418497 I have suggested to anyone to give me the approximations of the minima for the laguerre polynomial, but someone have suggested the zero Laguerre polynomials. But i want the general formula for the approximation of the local minima for the generalized Laguerre polynomial. Since, we have the formula given in the Book "Orthogonal polynomials-Szego 1939" in page 240: $$Max_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{\alpha/2-1/12} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ My question is: Can the local minima of the Laguerre polynomial be written as the following $$Min_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{k} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$ where $-1 \leq k \leq \alpha/2-1/12$? If so why i have seen that because for that case $n^{k} \leq n^{\alpha/2}$. But i want someone to suggest to me a demonstration if that approximation is true? If not can we deduce another approximation?