Some Properties of Generalized Gegenbauer Matrix Polynomials

Table 1

List of known Hermite matrix and Laguerre matrix polynomials.

S. no.	Polynomials	Series definition	Generating functions
I	2-variable Hermite matrix	𝐻𝑛𝑥,𝑦,𝐴=𝑛!∑^[𝑛/2]𝑘=0(−1)^𝑘𝑦^𝑘(𝑥√2𝐴)^𝑛−2𝑘𝑘!(𝑛−2𝑘)!	exp𝑥𝑡√2𝐴−𝑦𝑡^2𝐼=∑^∞𝑛=0𝐻𝑛(𝑥,𝑦,𝐴)𝑡^𝑛𝑛!
	polynomials (2VHMaP)
	𝐻𝑛(𝑥,𝑦,𝐴) [2]
II	2-index 2-variable Hermite	𝐻𝑛,𝑚(𝑥,𝑦,𝐴)=𝑛!∑^[𝑛/𝑚]𝑘=0(−1)^𝑘𝑦^𝑘(𝑥√𝑚𝐴)^𝑛−𝑚𝑘𝑘!(𝑛−𝑚𝑘)!	exp𝑥𝑡√𝑚𝐴−𝑦𝑡^𝑚𝐼=∑^∞𝑛=0𝐻𝑛,𝑚(𝑥,𝑦,𝐴)𝑡^𝑛𝑛!
	matrix polynomials (2I2VHMaP)
	𝐻𝑛,𝑚(𝑥,𝑦,𝐴) [7]
III	3-index 3-variable Hermite	𝐻𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴)=𝑛!∑^[𝑛/𝑠]∑𝑘=0^[(𝑛−𝑠𝑘)/𝑚]𝑟=0(−1)^𝑟𝑧^𝑘𝑦^𝑟(𝑥√𝑚𝐴)^𝑛−𝑠𝑘−𝑚𝑟𝑘!𝑟!(𝑛−𝑠𝑘−𝑚𝑟)!	exp𝑥𝑡√𝑚𝐴−𝑦𝑡^𝑚𝐼+𝑧𝑡^𝑠𝐼=∑^∞𝑛=0𝐻𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴)𝑡^𝑛𝑛!
	matrix polynomials (3I3VHMaP)
	𝐻𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴) [8]
IV	2-variable Laguerre	𝐿𝑛^(𝐴,𝜆)(𝑥,𝑦)=∑^𝑛𝑘=0(−1)^𝑘(𝜆𝑥)^𝑘𝑦^𝑛−𝑘𝑘!(𝑛−𝑘)!(𝐴+𝐼)𝑛((𝐴+𝐼)𝑘)^−1	(1−𝑦𝑡)^−(𝐴+𝐼)exp−𝜆𝑥𝑡=1−𝑦𝑡^∞𝑛∑=0𝐿𝑛^(𝐴,𝜆)(𝑥,𝑦)𝑡^𝑛
	matrix polynomials (2VLMaP)
	𝐿𝑛^(𝐴,𝜆)(𝑥,𝑦) [9]
V	2-variable modified Laguerre	𝑓𝑛^(𝐴,𝜆)(𝑥,𝑦)=∑^𝑛𝑘=0(𝐴)𝑛−𝑘(𝜆𝑥)^𝑘𝑦^𝑛−𝑘𝑘!(𝑛−𝑘)!	(1−𝑦𝑡)^−𝐴exp(𝜆𝑥𝑡)=∑^∞𝑛=0𝑓𝑛^(𝐴,𝜆)(𝑥,𝑦)𝑡^𝑛
	matrix polynomials (2VMLMaP)
	𝑓𝑛^(𝐴,𝜆)(𝑥,𝑦) [9]