General-Appell Polynomials within the Context of Monomiality Principle

Table 2

List of some Gould-Hopper-Appell polynomials _𝐻𝐴_𝑛^(𝑚)(𝑥,𝑦).

S. No.	𝑔(𝑡); 𝐴(𝑡)	𝑀𝐻^(𝑚)𝐴; 𝑃𝐻^(𝑚)𝐴	Generating functions	Polynomials
(I)	𝑒^𝑡−1𝑡; 𝑡(𝑒^𝑡−1)	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+1−𝐷𝑥𝑒^𝐷𝑥−1𝐷𝑥𝑒^𝐷𝑥−1; 𝐷𝑥	𝑡(𝑒^𝑡)𝑒−1^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐵𝑛^(𝑚)𝑡𝑥,𝑦^𝑛𝑛!	The Gould-Hopper-Bernoulli polynomials
(II)	𝑒^𝑡+12; 2(𝑒^𝑡+1)	𝑥+𝑚𝑦𝐷𝑥^𝑚−1−𝑒^𝐷𝑥𝑒^𝐷𝑥+1; 𝐷𝑥	2(𝑒^𝑡)𝑒+1^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐸𝑛^(𝑚)𝑡𝑥,𝑦^𝑛𝑛!	The Gould Hopper-Euler; polynomials
(III)	𝑒^𝑡−1^𝛼𝑡^𝛼; 𝑡^𝛼(𝑒^𝑡−1)^𝛼	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+𝛼1−𝐷𝑥𝑒^𝐷𝑥−1𝐷𝑥𝑒^𝐷𝑥−1; 𝐷𝑥	𝑡^𝛼(𝑒^𝑡−1)^𝛼𝑒^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐵𝑛^(𝑚,𝛼)𝑡𝑥,𝑦^𝑛𝑛!	The Gould-Hopper-generalized Bernoulli polynomials
(IV)	𝑒^𝑡+1^𝛼2^𝛼; 2^𝛼(𝑒^𝑡+1)^𝛼	𝑥+𝑚𝑦𝐷𝑥^𝑚−1−𝛼𝑒^𝐷𝑥𝑒^𝐷𝑥+1; 𝐷𝑥	2^𝛼(𝑒^𝑡+1)^𝛼𝑒^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐸𝑛^(𝑚,𝛼)𝑡𝑥,𝑦^𝑛𝑛!	The Gould-Hopper-generalized; Euler polynomials
(V)	^𝑠∏ℎ=1𝑒^𝛼ℎ𝑡−1𝛼ℎ𝑡^𝑠;	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+𝑚𝐷𝑥^−1−∑^𝑚𝑟=1𝛼𝑟𝑒^𝛼𝑟𝐷𝑥𝑒^𝛼𝑟𝐷𝑥−1; 𝐷𝑥	𝛼1⋯𝛼𝑠𝑡^𝑠(𝑒^𝛼1𝑡−1)⋯(𝑒^𝛼𝑠𝑡)𝑒−1^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐵𝑛^(𝑚,𝑠)(𝑥,𝑦∣𝛼1,…,𝛼𝑠)𝑡^𝑛𝑛!	The Gould-Hopper-generalized Bernoulli polynomials of order 𝑠
	^𝑠∏ℎ=1𝛼ℎ(𝑒^𝛼ℎ𝑡)𝑡−1^𝑠
(VI)	^𝑠∏ℎ=1𝑒^𝛼ℎ𝑡+12^𝑠;	𝑥+𝑚𝑦𝐷𝑥^𝑚−1−∑^𝑚𝑟=1𝛼𝑟𝑒^𝛼𝑟𝐷𝑥; 𝐷𝑥	2^𝑠(𝑒^𝛼1𝑡+1)⋯(𝑒^𝛼𝑠𝑡)𝑒+1^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐸𝑛^(𝑚,𝑠)(𝑥,𝑦∣𝛼1,…,𝛼𝑠)𝑡^𝑛𝑛!	The Gould-Hopper-generalized; Euler polynomials of order 𝑠
	^𝑠∏ℎ=12^𝑠(𝑒^𝛼ℎ𝑡+1)
(VII)	𝑒^𝑡−∑^𝑠−1ℎ=0𝑡^ℎ/ℎ!𝑡^𝑠;	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+𝑚𝐷𝑥^−1𝐷−1−^𝑥𝑚−1/(𝑚−1)!𝑒^𝐷𝑥−∑^𝑚−1ℎ=0𝑡^ℎ/ℎ!; 𝐷𝑥	𝑡^𝑠𝑒^𝑡−∑^𝑠−1ℎ=0𝑡^ℎ𝑒/ℎ!^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐵𝑛^(𝑚,[𝑠−1])𝑡𝑥,𝑦^𝑛𝑛!	The Gould-Hopper new; generalized Bernoulli polynomials
	𝑡^𝑠𝑒^𝑡−∑^𝑠−1ℎ=0𝑡^ℎ/ℎ!
(VIII)	𝑡𝜆𝑒^𝑡−1^−𝛼; 𝑡𝜆𝑒^𝑡−1^𝛼	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+𝜆𝑒^𝐷𝑥𝛼−𝐷𝑥−𝛼𝐷𝑥𝜆𝑒^𝐷𝑥−1; 𝐷𝑥	𝑡𝜆𝑒^𝑡−1^𝛼𝑒^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝔅𝑛^(𝑚,𝛼)𝑡𝑥,𝑦;𝜆^𝑛𝑛!	The Gould-Hopper-Apostol-Bernoulli polynomials of order 𝛼
(IX)	𝜆𝑒^𝑡−1𝑡; 𝑡𝜆𝑒^𝑡−1	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+𝜆𝑒^𝐷𝑥1−𝐷𝑥−1𝐷𝑥𝜆𝑒^𝐷𝑥−1; 𝐷𝑥	𝑡𝜆𝑒^𝑡𝑒−1^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝔅𝑛^(𝑚)𝑡𝑥,𝑦;𝜆^𝑛𝑛!	The Gould-Hopper-Apostol-Bernoulli polynomials
(X)	2𝜆𝑒^𝑡+1^−𝛼; 2𝜆𝑒^𝑡+1^𝛼	𝑥+𝑚𝑦𝐷𝑥^𝑚−1−𝜆𝛼𝑒^𝐷𝑥𝜆𝑒^𝐷𝑥+1; 𝐷𝑥	2𝜆𝑒^𝑡+1^𝛼𝑒^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0ℰ𝑛^(𝑚,𝛼)𝑡𝑥,𝑦;𝜆^𝑛𝑛!,𝑡+log𝜆<𝜋;1^𝛼∶=1	The Gould-Hopper-Apostol-Euler polynomials of order 𝛼
(XI)	𝜆𝑒^𝑡+12; 2𝜆𝑒^𝑡+1	𝑥+𝑚𝑦𝐷𝑥^𝑚−1−𝜆𝑒^𝐷𝑥𝜆𝑒^𝐷𝑥+1; 𝐷𝑥	2𝜆𝑒^𝑡𝑒+1^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0ℰ𝑛^(𝑚)𝑡𝑥,𝑦;𝜆^𝑛,𝑛!𝑡+log𝜆<𝜋	The Gould-Hopper-Apostol-Euler polynomials
(XII)	𝑒^−(∑𝑟+1ℎ=0𝜉ℎ𝑡ℎ); 𝑒^(∑𝑟+1ℎ=0𝜉ℎ𝑡ℎ); 𝜉𝑟+1≠0	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+∑^𝑟+1ℎ=1ℎ𝜉ℎ𝐷𝑥^ℎ−1; 𝐷𝑥	𝑒^(∑𝑟+1ℎ=0𝜉ℎ𝑡ℎ+𝑥𝑡+𝑦𝑡𝑚)=∑^∞𝐻𝑛=0𝐴𝑛^(𝑚)𝑡𝑥,𝑦^𝑛𝑛!	The Gould-Hopper-generalized Gould-Hopper polynomials (for 𝑟=1, the Gould-Hopper-Hermite polynomials 𝐻𝐻𝑛^(𝑚)𝑥,𝑦 and for 𝑟=2, the Gould-Hopper classical 2-orthogonal polynomials)
(XIII)	(1−𝑡)^𝑠+1; 1(1−𝑡)^𝑠+1	𝑥+𝑚𝑦𝐷𝑥^𝑚−1−𝑚+11−𝐷𝑥; 𝐷𝑥	1(1−𝑡)^𝑠+1𝑒^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐺𝑛^(𝑚,𝑠)𝑥,𝑦𝑡^𝑛	The Gould-Hopper-Miller-Lee polynomials (for 𝑠=0, the Gould-Hopper-truncated exponential polynomials 𝐻𝑒^𝑚𝑛(𝑥,𝑦) and for 𝑠=𝛽−1 Gould-Hopper-modified Laguerre polynomials 𝐻𝑓𝑛^(𝑚,𝛽)(𝑥,𝑦))
(XIV)	𝑒^𝑡+12𝑡; 2𝑡(𝑒^𝑡+1)	𝑥+𝑚𝑦𝐷𝑥^𝑚−1+𝑒^𝐷𝑥1−𝐷𝑥+1𝑒^𝐷𝑥+1; 𝐷𝑥	2𝑡(𝑒^𝑡)𝑒+1^𝑥𝑡+𝑦𝑡𝑚=∑^∞𝐻𝑛=0𝐺𝑛^(𝑚)𝑡𝑥,𝑦^𝑛𝑛!	The Gould-Hopper-Genocchi polynomials